[Theorem~\ref{thm:mainthm} \& Corollary~\ref{cor:spec}]
\label{thm:1}
Let $G$ be simply connected. The coordinate ring of the Vinberg monoid $\textnormal{Env } G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible.

The map $\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.

\label{sec:intro}
In this paper, all varieties are defined over $\BC$.

\subsection{The Vinberg monoid}
\label{sec:intro:monoid}
An \emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \cite{Put84} and Renner \cite{Ren85}, and it has attracted many attentions since then.

In the work of \cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \emph{Vinberg monoid}. The Vinberg monoid $\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\textnormal{Env } G$ by a base change.

Moreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map
\[
\pi: \textnormal{Env } G\longrightarrow \mathbb{C}^r
\]
onto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\pi^{-1}(\mathbf{0})$ is the \emph{asymptotic semigroup} considered by Vinberg \cite{Vin95a}.

The Vinberg monoid together with the map $\pi$ have found significant applications in recent developments of the geometric representation theory, see \citelist{\cite{BNS16}\cite{FKM20}} for example.

\subsection{Cluster algebras}
\label{sec:intro:cluster}

The theory of cluster algebras was developed by Fomin and Zelevinsky \cite{FZ02} to study total positivity and dual canonical bases in semisimple groups. The coordinate rings of many varieties related to algebraic groups have shown to be (upper) cluster algebras (See \citelist{\cite{BFZ05}\cite{Wil13}\cite{Sco06}\cite{CGG+25}\cite{GLSS}} for example.).

In contrast, cluster algebra structures on algebraic monoids have received little attention. It seems that the only nontrivial example is the cluster algebra structure on the space of $n\times n$ matrices from \cite{FWZ20}*{Theorem~6.6.1}.

The existence of a cluster structure on a variety has a wide range of consequences. For example, it provides a total positivity structure, Poisson structure \cite{GSV10}, and a cluster theoretic canonical basis on its coordinate ring \cite{GHKK18}.

In this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \emph{partially compactified (upper) cluster algebras} from \cite{GHKK18} to highlight this choice of convention.

\subsection{Main results}
\label{sec:intro:main}

The main result of this paper is the follows.

\subsection{The Vinberg monoid}
\label{sec:intro:monoid}
An \emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \cite{Put84} and Renner \cite{Ren85}, and it has attracted many attentions since then.

In the work of \cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \emph{Vinberg monoid}. The Vinberg monoid $\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\textnormal{Env } G$ by a base change.

Moreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map
\[
\pi: \textnormal{Env } G\longrightarrow \mathbb{C}^r
\]
onto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\pi^{-1}(\mathbf{0})$ is the \emph{asymptotic semigroup} considered by Vinberg \cite{Vin95a}.

In this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \emph{partially compactified (upper) cluster algebras} from \cite{GHKK18} to highlight this choice of convention.

\subsection{Main results}
\label{sec:intro:main}

The main result of this paper is the follows.

\begin{cor}\label{cor:spec}
The diagram
\begin{equation}
    \begin{tikzcd}
    \Env G \arrow[dr,"\pi"'] \arrow[rr,"\sim"] && \Spec \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[dl,"\pi^{I'}"] \\ & \BC^I
    \end{tikzcd}
\end{equation}
commutes.
\end{cor}

\begin{theorem}
\label{thm:mainthm}
Let $\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\Sigma=\wt{J}_\fro\backslash I'$. The isomorphism $\Psi_\bfi$ restricts to isomorphisms
\[
\overline{\CU}^\Sigma(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[G\x ^Z T],\qquad \overline{\CU}(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[\Env G].
\]
In other words, we have the following commutative diagram
\begin{equation}\label{eq:cr}
    \begin{tikzcd}
        & \CU(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] & \BC[G^{w_0,w_0}\x ^Z T] \\
        & \overline{\CU}^\Sigma(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] \arrow[u,hook] & \BC[G\x^Z T] \arrow[u,hook] \\
        & \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[u,hook] \arrow[r,"\sim"] & \BC[\Env G] \arrow[u,hook]
    \end{tikzcd}
\end{equation}
\end{theorem}

The main result of this paper is the follows.

Combining with the universal property of the Vinberg monoid and the coefficient specialization of cluster algebras, Theorem \ref{thm:1} allows us to construct cluster algebra structures on a large family of reductive monoids.

\begin{thmintro}
[Theorem~\ref{thm:affineAbelianCluster}]
\label{thm:3}
Let $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\mathbb{C}^k$ (See Section \ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.
\end{thmintro}

Let us remark that the assumptions on the reductive monoid $M$ are reasonable. Any set of frozen variables comes with a map $M\to \BC^k$. This should be compatible with the abelianization for a natural cluster structure on $M$. On the other hand, our assumptions on $M$ is flexible enough to cover interesting examples. For example, they include the case where $M$ is the monoid of $n\times n$ matrices. In this case, Theorem \ref{thm:3} recovers the well-known cluster structures, presented in \cite{FWZ20}*{Theorem~6.6.1}.

\begin{thmintro}[Theorem~\ref{thm:reductiveMonoidCluster}]
\label{thm:2}
Let $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.
\end{thmintro}

\begin{theorem}
\label{thm:reductiveMonoidCluster}
Let $H$ be a connected reductive group with simply connected derived group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra $\overline{\mathcal{U}}^\Sigma$, whose further partially compactification $\overline{\mathcal{U}}$ along all frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.
\end{theorem}

\begin{theorem}\label{thm:affineAbelianCluster}
Let $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ isomorphic to an affine space $\mathbb{C}^n$. Then the coordinate ring $\BC[M]$ is a partially compactified upper cluster algebra with all frozen variables not invertible.
\end{theorem}

\begin{example}
Let $M=M_n$ be the monoid of $n\times n$ matrices, so $G(M)=GL_n$ and $G_M=SL_n$. In this case, the subalgebra $\BC[M]^{G_M\times G_M}$ of invariant functions is isomorphic to the polynomial algebra in one variable, generated by the determinant. Hence the abelianization $A(M)$ is isomorphic to the affine line. Theorem~\ref{thm:affineAbelianCluster} gives a partially compactified upper cluster algebra structure on $\BC[M_n]$, which coincides with the cluster structure in \cite{FWZ20}*{Theorem 6.6.1}.
\end{example}

\begin{cor}\label{cor:spec}
The diagram
\begin{equation}
    \begin{tikzcd}
    \Env G \arrow[dr,"\pi"'] \arrow[rr,"\sim"] && \Spec \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[dl,"\pi^{I'}"] \\ & \BC^I
    \end{tikzcd}
\end{equation}
commutes.
\end{cor}

\begin{thmintro}[Theorem~\ref{thm:mainthm} \& Corollary~\ref{cor:spec}]
\label{thm:1}
Let $G$ be simply connected. The coordinate ring of the Vinberg monoid $\Env G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible.

The map $\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.
\end{thmintro}

\begin{theorem}
\label{thm:mainthm}
Let $\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\Sigma=\wt{J}_\fro\backslash I'$. The isomorphism $\Psi_\bfi$ restricts to isomorphisms
\[
\overline{\CU}^\Sigma(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[G\x ^Z T],\qquad \overline{\CU}(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[\Env G].
\]
In other words, we have the following commutative diagram
\begin{equation}\label{eq:cr}
    \begin{tikzcd}
        & \CU(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] & \BC[G^{w_0,w_0}\x ^Z T] \\
        & \overline{\CU}^\Sigma(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] \arrow[u,hook] & \BC[G\x^Z T] \arrow[u,hook] \\
        & \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[u,hook] \arrow[r,"\sim"] & \BC[\Env G] \arrow[u,hook]
    \end{tikzcd}
\end{equation}
\end{theorem}

Combining with the universal property of the Vinberg monoid and the coefficient specialization of cluster algebras, Theorem \ref{thm:1} allows us to construct cluster algebra structures on a large family of reductive monoids.

\begin{thmintro}
[Theorem~\ref{thm:affineAbelianCluster}]
\label{thm:3}
Let $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\mathbb{C}^k$ (See Section \ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.
\end{thmintro}

Let us remark that the assumptions on the reductive monoid $M$ are reasonable. Any set of frozen variables comes with a map $M\to \BC^k$. This should be compatible with the abelianization for a natural cluster structure on $M$. On the other hand, our assumptions on $M$ is flexible enough to cover interesting examples. For example, they include the case where $M$ is the monoid of $n\times n$ matrices. In this case, Theorem \ref{thm:3} recovers the well-known cluster structures, presented in \cite{FWZ20}*{Theorem~6.6.1}.

By inverting the appropriate frozen variables, we obtain cluster algebra structures on any connected reductive group.

\begin{thmintro}[Theorem~\ref{thm:reductiveMonoidCluster}]
\label{thm:2}
Let $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.
\end{thmintro}

In the case when $H$ is semisimple, Theorem \ref{thm:2} was obtained by Qin and Yakimov \cite{QY25}. Independently, Oya \cite{Oya25} proved the same result under the further condition that $H$ is not of type $F_4$. In the semisimple case, there are no additional invertible frozen variables. This is also reflected by the fact that a reductive monoid with semisimple unit group is necessarily a group.

\subsection{Strategy of the proof}
\label{sec:intro:strategy}

The first key ingredient of the proof of Theorem~\ref{thm:1} is the construction of a framed group $\widetilde{G}$ of $G$. Let $I$ be the set of simple roots of $G$ with $|I|=r$. We define $\widetilde{I}$ by adding $r$ vertices labeled $i'$ and a simple edge joining $i$ and $i'$ for each $i\in I$. The framed group $\widetilde{G}$ is defined to be the corresponding Kac--Moody group. We then construct an isomorphism between the unit group of $\textnormal{Env } G$ to the standard Levi subgroup of $\widetilde{G}$ associated with $I$. Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \cite{Wil13} (See also \cite{SW21}), we obtain a cluster structure on an open part $(\textnormal{Env } G)^\circ$ of $\textnormal{Env } G$.

The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\textnormal{Env } G-(\textnormal{Env } G)^\circ$. This is achieved by applying the string parametrization of the dual canonical basis.

We note that the technique of extending root data was inspired by the thickening map introduced by Bao and He \cite{BH24}. Similar ideas appeared in the recent works \citelist{\cite{Qin25}\cite{FH25}} as well.

\subsection{Future problems}
\label{sec:intro:future}

Let us close the introduction by mentioning some future directions.

Firstly, reductive monoids admit standard Poisson structures and quantization \cite{BS25}. We expect that all the results in the current paper can be extended to the quantum setting.

Secondly, we focus only on upper cluster algebras in this paper. For simply-connected simple group $G$ not of type $F_4$, Oya \cite{Oya25} proved that the upper cluster algebra structure on $\mathbb{C}[G]$ coincides with the ordinary cluster algebra in the sense of Fomin and Zelevinsky. A quantum analogue of this coincidence was recently established by Oya, Qin, and Yakimov \cite{OQY25}. It is therefore a natural question whether the upper cluster algebras studied in this paper also coincide with ordinary cluster algebras.

Thirdly, the Vinberg monoid $\textnormal{Env } G$ is isomorphic to the total coordinate ring of the wonderful compactification $\mathbf{X}$ of $G_{ad}=G/Z$. It would be interesting to explore the connections between the cluster algebra structure on $\textnormal{Env } G$ developed in the current paper with the total positivity on $\mathbf{X}$ studied by He \cite{He04}.

\vspace{.2cm}

\noindent {\bf Acknowledgment: } We are grateful to Huanchen Bao for valuable discussions and suggestions during the writing of this paper. JS is grateful to Fan Qin for helpful discussions at the early stage of the paper. JS is supported by the Glorious Sun Charity Fund.

The post-theorem introduction contains a proof strategy for Theorem~\ref{thm:1}:

1. **Construct a framed group and relate it to $\textnormal{Env }G$.** “The first key ingredient of the proof of Theorem~\ref{thm:1} is the construction of a framed group $\widetilde{G}$ of $G$.” With $I$ the simple roots, define $\widetilde{I}$ by adding vertices $i'$ and an edge between $i$ and $i'$ for each $i\in I$, and let $\widetilde{G}$ be the associated Kac--Moody group. Then “we construct an isomorphism between the unit group of $\textnormal{Env } G$ to the standard Levi subgroup of $\widetilde{G}$ associated with $I$.”

2. **Import cluster structures from Kac--Moody double Bruhat cells to an open part of $\textnormal{Env }G$.** “Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \cite{Wil13} (See also \cite{SW21}), we obtain a cluster structure on an open part $(\textnormal{Env } G)^\circ$ of $\textnormal{Env } G$.”

3. **Compare valuations to control the boundary/frozen variables via canonical basis combinatorics.** “The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\textnormal{Env } G-(\textnormal{Env } G)^\circ$.” This comparison “is achieved by applying the string parametrization of the dual canonical basis.”

(Additional remark in the strategy section: “the technique of extending root data was inspired by the thickening map introduced by Bao and He \cite{BH24}.”)

The post-theorem introduction contains a proof strategy for the main theorem:

1. **Construct a framed group and relate it to $\textnormal{Env }G$.** “The first key ingredient of the proof of the main theorem is the construction of a framed group $\widetilde{G}$ of $G$.” With $I$ the simple roots, define $\widetilde{I}$ by adding vertices $i'$ and an edge between $i$ and $i'$ for each $i\in I$, and let $\widetilde{G}$ be the associated Kac--Moody group. Then “we construct an isomorphism between the unit group of $\textnormal{Env } G$ to the standard Levi subgroup of $\widetilde{G}$ associated with $I$.”

2. **Import cluster structures from Kac--Moody double Bruhat cells to an open part of $\textnormal{Env }G$.** “Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \cite{Wil13} (See also \cite{SW21}), we obtain a cluster structure on an open part $(\textnormal{Env } G)^\circ$ of $\textnormal{Env } G$.”

3. **Compare valuations to control the boundary/frozen variables via canonical basis combinatorics.** “The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\textnormal{Env } G-(\textnormal{Env } G)^\circ$.” This comparison “is achieved by applying the string parametrization of the dual canonical basis.”

(Additional remark in the strategy section: “the technique of extending root data was inspired by the thickening map introduced by Bao and He \cite{BH24}.”)

[To prove the main theorem \& we first prove the following corollary. \begin{cor}\label{cor:spec}
The diagram
\begin{equation}
    \begin{tikzcd}
    \Env G \arrow[dr,"\pi"'] \arrow[rr,"\sim"] && \Spec \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[dl,"\pi^{I'}"] \\ & \BC^I
    \end{tikzcd}
\end{equation}
commutes.
\end{cor}]
\label{thm:1}
Let $G$ be simply connected. The coordinate ring of the Vinberg monoid $\textnormal{Env } G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible.

The map $\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.,

Classification or Bijection
Asymptotic or Limit

Question: Let $G$ be a simply connected semisimple group, let $\operatorname{Env} G$ be its Vinberg monoid, and let $I$ index the simple roots, so that $\mathbb C^I\simeq \mathbb C^r$ with $r=\operatorname{rank}(G)$. The Vinberg monoid has a canonical flat morphism $\pi:\operatorname{Env} G\to \mathbb C^I$, whose special fiber over $\mathbf 0$ is the asymptotic semigroup. Let $\mathbf i$ be a double reduced word for $(w_0,w_0)$, let $\widetilde{\mathbf s}(\mathbf i)$ be the associated seed, and let $\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$ denote the partially compactified upper cluster algebra in which all frozen variables are not inverted. Let $\pi^{I'}: \operatorname{Spec}\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))\to \mathbb C^I$ be the morphism defined by the corresponding frozen-variable specialization. Which statement holds?

There is an isomorphism $\operatorname{Env} G\xrightarrow{\sim}\operatorname{Spec}\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, equivalently $\mathbb C[\operatorname{Env} G]\cong \overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, so the coordinate ring of $\operatorname{Env} G$ is a partially compactified upper cluster algebra with all frozen variables noninvertible. Under this isomorphism, the canonical map $\pi$ agrees with the specialization map $\pi^{I'}$, i.e. $\pi=\pi^{I'}\circ(\operatorname{Env} G\xrightarrow{\sim}\operatorname{Spec}\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i)))$. In particular, the asymptotic semigroup $\pi^{-1}(\mathbf 0)$ is obtained by specializing the relevant frozen variables to $0$.

Choices:
B. There is an isomorphism $\operatorname{Env} G\xrightarrow{\sim}\operatorname{Spec}\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, equivalently $\mathbb C[\operatorname{Env} G]\cong \overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, and under this isomorphism the canonical map $\pi$ agrees with the specialization map obtained by setting all frozen variables to $0$. In particular, every fiber of $\pi$ is obtained by specializing all frozen variables simultaneously, and the asymptotic semigroup is the generic fiber.
C. There is an isomorphism $\operatorname{Env} G\xrightarrow{\sim}\operatorname{Spec}\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, equivalently $\mathbb C[\operatorname{Env} G]\cong \overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$. In particular, the coordinate ring of $\operatorname{Env} G$ is a partially compactified upper cluster algebra with all frozen variables noninvertible.
D. For each double reduced word $\mathbf i$ for $(w_0,w_0)$, there is an isomorphism $\operatorname{Env} G\xrightarrow{\sim}\operatorname{Spec}\mathcal U(\widetilde{\mathbf s}(\mathbf i))$, equivalently $\mathbb C[\operatorname{Env} G]\cong \mathcal U(\widetilde{\mathbf s}(\mathbf i))$, where the frozen variables remain invertible. Under this isomorphism, the map $\pi$ is induced by a regular function on the open double Bruhat-cell chart and extends uniquely to all of $\operatorname{Env} G$.
E. There is an isomorphism $\operatorname{Env} G\xrightarrow{\sim}\operatorname{Spec}\overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, equivalently $\mathbb C[\operatorname{Env} G]\cong \overline{\mathcal U}(\widetilde{\mathbf s}(\mathbf i))$, but the canonical flat map $\pi:\operatorname{Env} G\to \mathbb C^I$ does not in general coincide with the frozen-variable specialization $\pi^{I'}$; rather, it agrees only after restricting to the open part corresponding to the unit group, while on the boundary divisor the two maps differ.

special-fiber-vs-generic-fiber and specialization of certain frozen variables

dropped the identification of $\pi$ with $\pi^{I'}$ and the asymptotic-semigroup specialization statement

extension from open part $(\operatorname{Env} G)^\circ$ to the full monoid by keeping frozen variables invertible

valuation comparison ensuring boundary compatibility of $\pi$ and frozen-variable specialization


In this paper we are interested in the following well-known conjecture of Hindman.

\begin{conj}[\cite{hindman.conjecture}]\label{hindman conjecture}
    Any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}.$
\end{conj}

Despite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.

Most relevant to the present paper, Moreira \cite{moreira2017monochromatic} has shown that any finite coloring of $\mathbb{N}$ contains monochromatic sets $\{x,xy,x+y\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \cite{alweiss2022monochromatic}.  The second author \cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\mathbb{N}$ into two colors, the second and the third authors  have shown \cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\mathbb{Q}$ and the first author has shown \cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\mathbb{Q}$.

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations.  Our main result is the following, which was previously only known in the case $k=1.$

\begin{conj}[\cite{hindman.conjecture}]\label{hindman conjecture}
 Any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+y\}.$
\end{conj}

Despite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.

Most relevant to the present paper, Moreira \cite{moreira2017monochromatic} has shown that any finite coloring of $\mathbb{N}$ contains monochromatic sets $\{x,xy,x+y\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \cite{alweiss2022monochromatic}. The second author \cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\mathbb{N}$ into two colors, the second and the third authors have shown \cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\mathbb{Q}$ and the first author has shown \cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\mathbb{Q}$.

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$

In the present paper we return to the two-color case analyzed in \cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$

In addition to proving Theorem \ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations. Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \cite{bowen.non.commuting}. The basic strategy used in the proof of Theorem \ref{t1} and the related results \cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:

\begin{theorem}\label{t2}
 For any $k\in \mathbb{N}$ and any $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ with $\{x,y,x^y, xy^i: i\leq k\}$ monochromatic.
\end{theorem}

\begin{theorem}\label{rational.moreira}
 Let $P\subseteq\pol$ be finite. For any finite coloring of $\mathbb{N}$ there exist $x,y\in\mathbb{N}$ and a color class $C$ such that $\{x,xy,x+p(y):p\in P\}\subseteq C$.
\end{theorem}

\begin{prop}\label{central*}
 For any finite set of polynomials $P\subseteq\pol$ and any finite coloring of $\mathbb{N}$ the set $$\{y: \exists x \textnormal{ with } \{x,xy,x+p(y):p\in P\} \textnormal{ monochromatic} \} $$

\begin{proof}[Second proof of Corollary \ref{all_syndetic}]
 Fix a finite set $P\subseteq \pol.$ By definition, for each $i$ there is a finite set $F_i\subset \mathbb{N}$ such that $S_i/F_i\supseteq \mathbb{N}.$ Let $k=\max_{i\leq r}|F_i|$ and consider the coloring $c:\mathbb{N}\rightarrow [k]^r$ where an $n\in\mathbb{N}$ is colored based on the $r$-tuple listing the minimal $f_i\in F_i$ such that $f_in\in S_i.$ Let $P'=\{p/f: p\in P, f\in \bigcup_{i\leq r}F_i\}$. By Theorem \ref{rational.moreira}, there are integers $x',y$ such that $S=\{x', x'y, x'+p'(y): p'\in P'\}$ is monochromatic according to the coloring $c.$ Now if $y\in S_i$, by the definition of $c$ there is an $f_i$ such that $f_iS\subseteq S_i.$ Consequently, setting $x=x'f_i$ is as desired.
\end{proof}

\begin{lemma}\label{all thick}
 Suppose $k\in\mathbb{N}$ and $\mathbb{N}=R\cup B$ with both $R$ and $B$ multiplicatively thick. Then one of the colors contains a set $\{x,y,xy,x+iy:i\leq k\}$.
\end{lemma}
\begin{proof}
Suppose for the sake of contradiction that the lemma is false.
 Without loss of generality assume that $R$ is additively piecewise syndetic. Choose $N\in\mathbb{N}$ large enough depending on $k$.

\begin{lemma}\label{log reduction}
 If any finite coloring of $\mathbb{N}$ contains a monochromatic set $\{x,y, x2^y, x+iy: i\leq k\}$ then any finite coloring contains a monochromatic set of the form $\{x,y,x^y,xy^i: i\leq k\}.$
\end{lemma}

\begin{theorem}\label{t1}
    For any $k\in \mathbb{N}$ and $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ such that $\{x,y,xy, x+iy: i\leq k\}$ is monochromatic.
\end{theorem}

In addition to proving Theorem \ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations.  Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \cite{bowen.non.commuting}.  The basic strategy used in the proof of Theorem \ref{t1} and the related results \cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:

\begin{enumerate}
    \item First, show that the result holds when the color of the step size $y$ is ignored.  In this case, we are interested in finding monochromatic sets $\{x,xy,x+iy: i\leq k\},$ which can be done either through Moreira's work \cite{moreira2017monochromatic} or the first author's \cite{alweiss2022monochromatic}.

    \item Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.

    \item Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well. \end{enumerate}

In order to illustrate this technique we give two proofs of Theorem \ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.  These are presented in the two subsections of Section \ref{section +}.

We also show that the same basic approach can be adapted to prove the following exponential version of the result.

\begin{theorem}\label{t2}
    For any $k\in \mathbb{N}$ and any $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ with $\{x,y,x^y, xy^i: i\leq k\}$ monochromatic.
\end{theorem}

The $k=1$ case of this result was proven by Sahasrabudhe \cite{sahasrabudhe2018monochromatic}, even for arbitrary finite colorings.  However, the result seems to be new for $k>1.$  Along the way, we also prove that any finite coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,x^y,xy^i: i\leq k\},$ which gives a new exponential version of Moreira's theorem.

The authors state that “the basic strategy used in the proof of Theorem \ref{t1} … is essentially as follows:”
\begin{enumerate}
\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\{x,xy,x+iy: i\leq k\}$, “which can be done either through Moreira's work … or the first author's …”.
\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”
\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”
\end{enumerate}
They further note that they “give two proofs of Theorem \ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”

The authors state that “the basic strategy used in the proof of the main theorem … is essentially as follows:”
\begin{enumerate}
\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\{x,xy,x+iy: i\leq k\}$, “which can be done either through Moreira's work … or the first author's …”.
\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”
\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”
\end{enumerate}
They further note that they “give two proofs of the main theorem based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”

\label{t1}
    For any $k\in \mathbb{N}$ and $2$-coloring of $\mathbb{N},$ there exist $x,y\in\mathbb{N}$ such that $\{x,y,xy, x+iy: i\leq k\}$ is monochromatic.,

Question: Let $k\in\mathbb{N}$, and let $\mathbb{N}$ be colored with two colors (equivalently, let $c:\mathbb{N}\to\{1,2\}$ be any 2-coloring). Which statement holds?

There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, xy,\, x+iy: i\le k\}$ have the same color; equivalently, $\{x,y,xy,x+y,x+2y,\dots,x+ky\}$ is monochromatic.

Choices:
B. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, xy,\, x+iy: i\le k\}$ have the same color; equivalently, $\{x,xy,x+y,x+2y,\dots,x+ky\}$ is monochromatic.
C. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, x+y,\, x+2y,\dots,x+ky\}$ have the same color.
D. For every color class $C$ of the 2-coloring, there exist $x,y\in\mathbb{N}$ such that $\{x,\, y,\, xy,\, x+iy: i\le k\}\subseteq C$.
E. There exist $x,y\in\mathbb{N}$ such that all elements of the set $\{x,\, y,\, x^y,\, xy^i: i\le k\}$ have the same color; equivalently, $\{x,y,x^y,xy,xy^2,\dots,xy^k\}$ is monochromatic.


\begin{thm}\label{thm: main}
Consider \eqref{eq: main} with random initial data \eqref{eq: rmain}. Let $c_{n}$ in \eqref{eq: rmain} be $\langle n\rangle^{-(\frac{1}{2}+\theta)},\theta>0$. Let $T_{\epsilon}\approx \epsilon^{-1}$. One has the following large deviation principle
\begin{equation}\label{eq: estimatemain}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon\ln \mathbb{P}(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}|c_{n}|^{2}}.
\end{equation}
\end{thm}

\subsection{Statement of results}
Consider cubic NLS on 1d torus $\mathbb{T}$,
\begin{equation}\label{eq: main}
\begin{cases}
iu_{t}+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_{0}(x)
\end{cases}
\end{equation}
here $\epsilon>0$ is a parameter.

We will focus on the random initial data of type
\begin{equation}\label{eq: rmain}
u^{\omega}_{0}(x)=\sum_{n}c_{n}g_{n}^{\omega}e^{inx}
\end{equation}
where $c_{n}=\<n\>^{-(\frac{1}{2}+\theta)}, \theta>0$, and $g_{n}^{\omega}$ be i.i.d standard complex Gaussian.\\

Note that $u^{\omega}_{0}\in L_{x}^{2}$ almost surely, and it is well known that all $L^{2}$ initial data generates a global flow, \cite{bourgain1993fourier}. Our main result is

\begin{equation}\label{eq: main}
\begin{cases}
iu_{t}+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_{0}(x)
\end{cases}
\end{equation}

We will focus on the random initial data of type
\begin{equation}\label{eq: rmain}
u^{\omega}_{0}(x)=\sum_{n}c_{n}g_{n}^{\omega}e^{inx}
\end{equation}
where $c_{n}=\<n\>^{-(\frac{1}{2}+\theta)}, \theta>0$, and $g_{n}^{\omega}$ be i.i.d standard complex Gaussian.\\

Both \cite{garrido2023large}, \cite{liang2025random}, and this work actually work for time scale for $c\epsilon^{-1}|\ln \epsilon|$, for some $c$ small but universal. Such improvement comes from standard improvement in Gronwall argument. In \cite{liang2025random}, they state the time scale as $O(\epsilon^{-1}|\ln \epsilon|)$, however, it seems hard to cover time scale for $C\epsilon^{-1}|\ln \epsilon|$ for $C$ large, see the line below (3.30) in \cite{liang2025random}.
\end{rem}

\begin{lem}\label{lem: modlinearldp}
Let $z_{0}>0$, $T_{\epsilon}=O(\epsilon^{-1})$ and $u_{app}$ be defined as \eqref{eq: uapp}, one has the large deviation principle,
\begin{equation}\label{eq: modlinearldp}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon \ln \mathbb{P}(\|u_{app}\|_{L_{t,x}^{\infty}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}c_{n}^{2}}
\end{equation}

and
\begin{lem}\label{lem: errorcontrol}
Let $u_{0}$ be as in \eqref{eq: u0}, and $u_{app}$ be as in \eqref{eq: uapp}. Let $u$ be the associated solution to \eqref{eq: main} with initial data $u_{0}$. Let $\theta>0$ be fixed. Let $T_{\epsilon}=O(\epsilon^{-1})$. Let $C_{0}\gg 1$. Then there exists $s>\frac{1}{2}$, and $\delta_{1}>0$, so that
\begin{equation}\label{eq: errorcontrol}
-\ln\mathbb{P}(\|u-u_{app}\|_{L_{t}^{\infty}H^{s}([0,T_{\epsilon}]\times \mathbb{T})}\geq \epsilon^{-\frac{1}{2}+\delta_{1}})\geq C_{0}\epsilon^{-1}
\end{equation}

By \cite[Lemma 4.2]{garrido2023large} or \cite[Lemma 4.2]{Tzvetkov2017quasi}, $\eta$ is a standard complex Gaussian if and only $\eta e^{it|\eta|^{2}}$ is a standard complex Gaussian. Our proof for Lemma \ref{lem: modlinearldp} actually yields to the same result\footnote{We will not use this result in the proof, and strictly speaking those two results, though same, are not equivalent, but we feel it is good to record this estimate for linear flow here.} for the linear free flow.
\begin{lem}\label{lem: ldpforlinearflow}
 Let $z_{0}>0$, $T_{\epsilon}=O(\epsilon^{-1})$ and $u_{0}$ be as in \eqref{eq: u0}, one has the large deviation principle
\begin{equation}\label{eq: linearldp}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon \ln \mathbb{P}(\|e^{it\Delta}u_{0}\|_{L_{t,x}^{\infty}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}c_{n}^{2}}
\end{equation}
\end{lem}

In this case, we we reduce \eqref{eq: crucialreduce} to proving that for all $\omega \in F_\epsilon - G_\epsilon$, one has
 \begin{equation}\label{eq: I-I-II}
 \sup_{0\leq \tau\leq t} \sum_{R_{n}(n_1,n_2,n_3)} \frac{|g_{n_1}g_{n_2}h_{n_3}|}{\<n_1\>^{\frac{1}{2}+\theta }\<n_2\>^{\frac{1}{2}+\theta }}
 \lesssim \<n\>^{-\theta + \delta_5} \epsilon ^{-3/2-3\delta_1}.
 \end{equation}
 Using Cauchy-Schwarz inequality and \eqref{eq: gepsic}, for all $\omega \in F_\epsilon - G_\epsilon$, we have
 \begin{equation}\label{eq: uniformI-I-II}
 \begin{aligned}
 &\sup_{0\leq \tau\leq t} \sum_{R_{n}(n_1,n_2,n_3)} \frac{|g_{n_1}g_{n_2}h_{n_3}|}{\<n_1\>^{\frac{1}{2}+\theta }\<n_2\>^{\frac{1}{2}+\theta }}\\
 &\lesssim \|h\|_{C_t[0,T]H^s} \left(\sum_{n_3} \<n_3\>^{-2s} \left|\sum_{R_{n_3,n}(n_1,n_2)} \frac{\epsilon ^{-1-2\delta_1 } }{\left\langle n_1 \right\rangle ^{1/2+ \theta -\delta_4 } \left\langle n_2 \right\rangle ^{1/2 + \theta -\delta_4}}\right|^2\right)^\frac{1}{2}\\
 &\lesssim \epsilon ^{-3/2 -3\delta_1 } \left(\sum_{n_3} \<n_3\>^{-2s} \left|\sum_{R_{n_3,n}(n_1,n_2)} \frac{1}{\left\langle n_1 \right\rangle ^{1/2+ \theta -\delta_4 } \left\langle n_2 \right\rangle ^{1/2 + \theta -\delta_4}}\right|^2\right)^\frac{1}{2},
 \end{aligned}
 \end{equation}
 where for fixed $n,n_3$, we denote $R_{n_3,n}(n_1,n_2) := \{(n_1,n_2)\in\mathbb{Z}^2:n=n_1-n_2+n_3,n_2\neq n_1,n_3\}$.
 Next, we aim to reduce \eqref{eq: I-I-II} to following
 \begin{equation}\label{eq: mainreductionI-I-II}
 \sum_{R_{n_3,n}(n_1,n_2)} \frac{1}{\left\langle n_1 \right\rangle ^{1/2+ \theta -\delta_4 } \left\langle n_2 \right\rangle ^{1/2 + \theta -\delta_4}}
 \lesssim \left\langle n-n_3 \right\rangle ^{-2\theta +2\delta_5} .
 \end{equation}
 If \eqref{eq: mainreductionI-I-II} holds, noting that
 \begin{equation*}
 \sum_{|n_3|\ll |n|} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \lesssim \<n\>^{-4\theta +4\delta_5}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_3|\gg |n|} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \sim \sum_{|n_3|\gg |n|} \<n_3\>^{-2s-4\theta +4\delta_5}
 \lesssim \<n\>^{-4\theta +4\delta_5}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_3|\sim |n|} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \sim \<n\>^{-2s} \sum_{|n_3|\sim |n|} \<n_3-n\>^{-4\theta +4\delta_5}
 \lesssim \<n\>^{1-2s-4\theta +4\delta_5},
 \end{equation*}
 then we have
 \begin{equation*}
 \sum_{n_3} \<n_3\>^{-2s} \left\langle n-n_3 \right\rangle ^{-4\theta +4\delta_5}
 \lesssim \<n\>^{-4\theta +4\delta_5},
 \end{equation*}
 which, along with \eqref{eq: mainreductionI-I-II} and \eqref{eq: uniformI-I-II}, yields \eqref{eq: I-I-II}. So we reduce \eqref{eq: I-I-II} to \eqref{eq: mainreductionI-I-II} if we take \fbox{$\delta_5< \theta $}. Similarly,
 noting that
 \begin{equation*}
 \sum_{|n_1|\ll |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta } \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \lesssim \left\langle n-n_3\right\rangle ^{\delta_4-\frac{1}{2}-\theta} \sum_{|n_1|\ll |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta }
 \lesssim \left\langle n-n_3\right\rangle ^{2\delta_4-2\theta}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_1|\gg |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta } \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \sim \sum_{|n_1|\gg |n-n_3|} \<n_1\>^{2\delta_4-1-2\theta }
 \lesssim \<n-n_3\>^{-2\theta +2\delta_4}
 \end{equation*}
 and
 \begin{equation*}
 \sum_{|n_1|\sim |n-n_3|} \<n_1\>^{\delta_4-\frac{1}{2}-\theta } \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \sim \left\langle n-n_3\right\rangle ^{\delta_4-\frac{1}{2}-\theta} \sum_{|n_1|\sim |n-n_3|} \left\langle n-n_3-n_1 \right\rangle ^{\delta_4-\frac{1}{2}-\theta}
 \lesssim \<n-n_3\>^{-2\theta +2\delta_4} ,
 \end{equation*}
 then we have \eqref{eq: mainreductionI-I-II}
 if we take \fbox{$\delta_5\leq \delta_4$}.

\begin{rem}
Upon finishing of this work, we notice a recent preprint of Liang-Wang, \cite{liang2025random},  which handles Theorem \ref{thm: main} in the case $\theta>\frac{1}{2}$. It should be noted that when $\theta>\frac{1}{2}$, the Fourier coefficient is in $l^{1}$, which forms an algebra. We handle all $\theta>0$.\\

Both \cite{garrido2023large}, \cite{liang2025random}, and this work actually work for time scale for $c\epsilon^{-1}|\ln \epsilon|$, for some $c$ small but universal. Such improvement comes from standard improvement in Gronwall argument. In \cite{liang2025random}, they state the time scale as $O(\epsilon^{-1}|\ln \epsilon|)$, however, it seems hard to cover time scale for $C\epsilon^{-1}|\ln \epsilon|$ for $C$ large, see the line below (3.30) in \cite{liang2025random}.
\end{rem}

\subsection{Background and motivation}
This work is inspired by the pioneering work \cite{garrido2023large}. Motivated by rogue wave phenomena, Garrido, Grande, Kurianski and Staffilani study \eqref{eq: main}, with random initial data of form \eqref{eq: rmain}, but for $c_{n}=ae^{-b|n|}$. They essentially
\footnote{They state the result for pointwise in $t\leq T_{\epsilon}\sim \epsilon^{-1}$. We take this chance to cover $\|u\|_{L_{t,x}^{\infty}}$ rather than estimate pointwise in $t$.} obtained \eqref{eq: estimatemain} for such types of random initial data. It has been remarked in \cite[Remark 1.3]{garrido2023large}, \textit{finding optimal family of coefficients (in terms of decay)}, rather than considering exponential decay $c_{n}$, remains an interesting open question.

By combining random data type techniques in \cite{bourgain1994periodic,bourgain1996invariant} and some ideas in \cite{compaan2021pointwise} (which also goes back to Bourgain's 1990s seminal work), we are able to extend the result in \cite{garrido2023large} to coefficients with polynomial decay $\langle n\rangle^{-\frac{1}{2}-\theta}$, $\theta>0$. Such constraint is optimal in the sense when $
\theta\leq 0$, even the initial data will leave $L_{x}^{\infty}$. It remains an interesting question to go beyond the time scale $T_{\epsilon}$. We note that this $\epsilon^{-1}$ time scale (neglecting logarithm corrections) is in some sense critical and we refer to \cite[Remark 1.7]{garrido2023large} for more details.
We note that both current work and \cite{garrido2023large},  can cover time scale of form $c\frac{|\ln \epsilon|}{\epsilon}$, where $c$ is some small universal number.\\

The overall proof scheme of current article follows \cite{garrido2023large}, which contains a large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part, (there is some subtle part here since one needs to first slightly perturb the free linear flow, we will discuss it later).\\

 Our main contributions in this work are
\begin{itemize}
\item We find a rather straightforward proof for the large deviation principle for the linear flow, which simplifies the arguments in \cite{garrido2023large}, and more importantly, extends to random initial data of form $\sum_{n}\frac{g^\omega _n}{\<n\>^{\frac{1}{2}+\theta}}e^{inx}$, $\forall \theta>0$. Our techniques for this part are crucial for our further analysis in the nonlinear smoothing part.
\item We adapt the $X^{s,b}$ analysis in \cite{bourgain1994periodic,bourgain1996invariant} in the setting of long time analysis of \cite{garrido2023large}, which gives the desired nonlinear smoothing\footnote{On one hand, we will use the $X^{s,b}$ analysis in the local random data theory as a black box;  on the other hand, the long time analysis we use to get nonlinear smoothing does explicitly use computations of $X^{s,b}$ type, but without going to the explicit form of $X^{s,b}$ analysis.}.
\end{itemize}

We end this session by pointing out ever since the seminal work \cite{bourgain1994periodic,bourgain1996invariant}, random initial data theory in nonlinear dispersive PDEs has become a very active research field. Many researchers contribute to this field and it is impossible to survey the field here. We simply mention that  in the perspective of  regularity in NLS model, the recent breakthrough \cite{deng2022random} gives a quite complete answer. We refer to \cite{deng2022random}, \cite{garrido2023large} and reference therein for more related reference.

\subsection{Notation}
Many terms $g_{n}^{\omega}, u^{\omega}, u_{0}^{\omega}$ are random, and thus depend on $\omega$. For simplicity, we will hide the implicit $\omega$ and simply denote them as $g_{n}, u,u_{0}$.\\

We introduce $\NNN_{1}, \NNN_{2}$ as
\begin{equation}\label{eq: n1}
\NNN_{1}(f_{1},f_{2},f_{3})=\sum_{n_{2}\neq n_{1}, n_{3}}\hat{f}_{1}(n_{1})\overline{\hat{f}_{2}}(n_{2})\hat{f}_{3}(n_{3})e^{i(n_{1}-n_{2}+n_{3})x},
\end{equation}
\begin{equation}\label{eq: n_{2}}
\NNN_{2}(f_{1},f_{2},f_{3})=\sum_{n}\hat{f}_{1}(n)\overline{\hat{f}_{2}}(n)\hat{f}_{3}(n)e^{inx}
\end{equation}
and one has
\begin{equation}\label{eq: algebra}
|f|^{2}f-2\frac{1}{2\pi}(\int |f|^{2})f=\NNN_{1}(f,f,f)-\NNN_{2}(f,f,f)
\end{equation}
We sometimes short $\NNN_{i}(f,f,f)$ as $\NNN_{i}(f)$.

The proof of Theorem~\ref{thm: main} is described as following the scheme of \cite{garrido2023large}: it “contains a large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” with “some subtle part here since one needs to first slightly perturb the free linear flow.” The authors then emphasize two main components: (i) “a rather straightforward proof for the large deviation principle for the linear flow,” which “simplifies the arguments in \cite{garrido2023large}” and extends to initial data $\sum_{n}\frac{g^\omega _n}{\<n\>^{\frac{1}{2}+\theta}}e^{inx}$ for all $\theta>0$; and (ii) adapting “the $X^{s,b}$ analysis in \cite{bourgain1994periodic,bourgain1996invariant} in the setting of long time analysis of \cite{garrido2023large}, which gives the desired nonlinear smoothing” for the Duhamel part.

The proof of the main theorem is described as following the scheme of Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” with “some subtle part here since one needs to first slightly perturb the free linear flow.” The authors then emphasize two main components: (i) “a rather straightforward proof for the large deviation principle for the linear flow,” which “simplifies the arguments in Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,”” and extends to initial data $\sum_{n}\frac{g^\omega _n}{\<n\>^{\frac{1}{2}+\theta}}e^{inx}$ for all $\theta>0$; and (ii) adapting “the $X^{s,b}$ analysis in Bourgain, “Periodic nonlinear Schrödinger equation and invariant measures,” and Bourgain, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation,” in the setting of long time analysis of Garrido and Oh, “Large deviation principle for the (modified) linear flow and a smoothing estimate for the Duhamel part,” which gives the desired nonlinear smoothing” for the Duhamel part.,

\label{thm: main}
Consider \begin{equation}\label{eq: main}
\begin{cases}\niu_{t}+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_{0}(x)
\end{cases}
\end{equation} with random initial data \begin{equation}\label{eq: rmain}\nu^{\omega}_{0}(x)=\sum_{n}c_{n}g_{n}^{\omega}e^{inx}
\end{equation}. Let $c_{n}$ in the equation above be $\langle n\rangle^{-(\frac{1}{2}+\theta)},\theta>0$. Let $T_{\epsilon}\approx \epsilon^{-1}$. One has the following large deviation principle
\begin{equation}\label{eq: estimatemain}
\lim_{\epsilon\rightarrow 0^{+}}\epsilon\ln \mathbb{P}(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb{T})}>z_{0}\epsilon^{-\frac{1}{2}})=-\frac{z_{0}^{2}}{\sum_{n}|c_{n}|^{2}}.
\end{equation},

Question:
Let \(\theta>0\), \(\langle n\rangle=(1+|n|^{2})^{1/2}\), and for each \(\epsilon>0\) consider the cubic nonlinear Schr\"odinger equation on the one-dimensional torus \(\mathbb T\):
\[
\begin{cases}
iu_t+\Delta u=\epsilon^{2}|u|^{2}u,\\
u(0,x)=u_0^{\omega}(x),
\end{cases}
\]
with random initial data
\[
u_0^{\omega}(x)=\sum_{n\in\mathbb Z} c_n g_n^{\omega} e^{inx}, \qquad c_n=\langle n\rangle^{-(\frac12+\theta)},
\]
where \(\{g_n^{\omega}\}_{n\in\mathbb Z}\) are i.i.d. standard complex Gaussian random variables. Let \(u^{\omega}\) denote the corresponding solution, and let \(T_{\epsilon}\) be comparable to \(\epsilon^{-1}\). For a fixed \(z_0>0\), which large-deviation estimate holds for
\(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}\) as \(\epsilon\to 0^+\)?

\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sum_{n\in\mathbb Z}|c_n|^2}.
\]

Choices:
B.
\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sup_{n\in\mathbb Z}|c_n|^2}.
\]
C.
\[
\limsup_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
\le -\frac{z_0^2}{\sum_{n\in\mathbb Z}|c_n|^2}.
\]
D.
\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\epsilon^{-1/2}\Big)
= -\frac{z_0^2}{2\sum_{n\in\mathbb Z}|c_n|^2}.
\]
E.
\[
\lim_{\epsilon\to 0^+} \epsilon\,\ln \mathbb P\Big(\|u^{\omega}\|_{L^{\infty}_{t,x}([0,T_{\epsilon}]\times \mathbb T)}> z_0\Big)
= -\frac{z_0^2}{\sum_{n\in\mathbb Z}|c_n|^2}.
\]

Gaussian variance from full \ell^2 mass \sum_n |c_n|^2

drop the full limit identity to the corresponding upper large-deviation bound

exact rate constant inherited from the complex Gaussian tail

critical threshold scaling z_0\epsilon^{-1/2} for the supremum norm


\label{thm:backbone}
There exist constants $C_1,C_2>0$ such that for any $\varepsilon\in(0,\frac{1}{10})$,
\begin{equation}\label{eq:bb-proba}
\frac{C_1}{\log|\log\varepsilon|}\le\mathbbm{P}[\Bac_\varepsilon]\le\frac{C_2}{\log|\log\varepsilon|}.
\end{equation}

\begin{theorem}\label{thm:backbone}
There exist constants $C_1,C_2>0$ such that for any $\varepsilon\in(0,\frac{1}{10})$,
\begin{equation}\label{eq:bb-proba}
\frac{C_1}{\log|\log\varepsilon|}\le\P[\Bac_\varepsilon]\le\frac{C_2}{\log|\log\varepsilon|}.
\end{equation}
\end{theorem}

\begin{equation}\label{eq:def-backbone}
\Bac_\varepsilon=\{\exists\ \text{two disjoint subpaths on the trajectory } B[0,\tau_{\D}] \text{ joining } \varepsilon\S^1 \text{ and } \frac{1}{2}\S^1\}.
\end{equation}

\begin{aligned}
F(\varepsilon)&\le \frac{C_0C'}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-n\log2}+q_0^n\\
&\le  \frac{C_0C'}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-\frac{\log2}{|\log q_0|}(\log\log|\log\varepsilon|-\log C')}+\frac{C'}{\log|\log\varepsilon|}
\le\frac{C_2}{\log|\log\varepsilon|}
\end{aligned}
$$
for some positive constant $C_2$, as desired.
\end{proof}

\appendix
\section{Proof of Theorem~\ref{thm-cr-mod-relation}}
\label{appendix}

In this appendix, we provide the proof of Theorem~\ref{thm-cr-mod-relation}, which is implicit in~\cite{ARS-Annulus}. For convenience, we consider the upper half plane $\hH:=\{z\in\C:\text{Im}(z)>0\}$, and let $\sm$ be the restriction of the $\SLE_{8/3}$ loop measure on $\C$ to the loops that are contained in $\hH$ and surround $i$. According to the conformal restriction~\cite{werner-loops}, if $\Phi:\D\to\hH$ is a conformal map with $\Phi(0)=i$, then $\sm=\Phi_*\SLE_{8/3,\D}^{\lp}$. Therefore, to prove Theorem~\ref{thm-cr-mod-relation}, it suffices to show the following
\begin{theorem}
\label{thm-appendix}
There is a constant $C>0$ such that for any measurable function $f:\R_+\to\R_+$ and $\lambda\ge0$, we have
$$
\int f\left(\Mod(\hH\backslash\overline{D(\eta)})\right)\psi_\eta'(i)^\lambda\sm(d\eta)=
C\frac{\sqrt{12\lambda-1}}{\sinh\left(\frac{\pi}{3}\sqrt{12\lambda-1}\right)}\int_0^\infty e^{-(2\lambda-\frac{1}{6})\pi\tau}\eta(2i\tau)f(\tau)d \tau.
$$
Here $\psi_\eta$ is the conformal map from $\hH$ to $D(\eta)$ such that $\psi_\eta(i)=i$ and $\psi_\eta'(i)>0$.
\end{theorem}

The proof of Theorem~\ref{thm-appendix} is based on the SLE-coupled Liouville quantum gravity with parameter $\sqrt{\frac{8}{3}}$. In the following we fix $\gamma=\sqrt{\frac{8}{3}}$ and let $Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Let $\P_{\hH}$ be the law of the free boundary Gaussian free field (GFF) on $\hH$ with mean zero on the upper semi-circle $\S^1\cap\hH$.

\begin{definition}[Liouville field on $\hH$]
\label{def-LF-H}
Let $(h,c)$ be sampled from $\P_{\hH}\times[e^{-Qc}dc]$, and let $\phi(z)=h(z)-2Q\log|z|_++c$ (here $|z|_+:=\max\{|z|,1\}$). Denote the law of $\phi$ by $\LF_{\hH}$.

For $\alpha\in\R$, let $\LF_{\hH}^{(\alpha,i)}(d\phi):=\lim_{\varepsilon\to0}\varepsilon^{\frac{\alpha^2}{2}}e^{\alpha\phi_\varepsilon(i)}\LF_{\hH}(d\phi)$ (the limit exists in the vague topology, see~\cite[Lemma 2.2]{ARS-FZZ}). Here $\phi_\varepsilon(i)$ is the average of $\phi$ over the circle $\partial B_\varepsilon(i)$.
\end{definition}

For a sample $\phi$ from $\LF_{\hH}^{(\alpha,i)}$, we can define its boundary Gaussian multiplicative chaos (GMC) measure $\nu_{\phi}:=\lim\limits_{\varepsilon\to0}\varepsilon^{\frac{\gamma^2}{4}}e^{\frac{\gamma}{2}\phi_\varepsilon(x)}dx$, where $\phi_\varepsilon(x)$ is the average of $\phi$ over the semi-circle $\partial B_\varepsilon(x)\cap\hH$.
Then denote $\{\LF_{\hH}^{(\alpha,i)}(\ell)\}_{\ell>0}$ to be the disintegration of $\LF_{\hH}^{(\alpha,i)}$ over $\nu_\phi(\R)$; i.e.~$\LF_{\hH}^{(\alpha,i)}=\int_0^\infty\LF_{\hH}^{(\alpha,i)}(\ell)d\ell$. We refer readers to see e.g.~\cite[Section 2.2]{ARS-FZZ} for further details. In particular, by~\cite[Lemma 2.7]{ARS-FZZ}, for $\alpha>\frac{\gamma}{2}$,
there are constants $C_{\alpha,\gamma}>0$ such that $|\LF_{\hH}^{(\alpha,i)}(\ell)|=C_{\alpha,\gamma}\ell^{\frac{2}{\gamma}(\alpha-Q)-1}$ for any $\ell>0$.

We also need the Liouville field on the annulus. For $\tau>0$, let $\mathcal{C}_\tau=[0,\tau]\times[0,1]/\mathord\sim$ be the horizontal cylinder with modulus $\tau$, where under $\sim$ we identify $(x,0)$ and $(x,1)$ for each $x\in[0,\tau]$. Let $\P_{\tau}$ be the free boundary GFF on $\mathcal{C}_\tau$
with zero mean on the circle $\{\frac{\tau}{2}\}\times[0,1]/\mathord\sim$.
\begin{definition}[{\cite[Definition 2.2]{ARS-Annulus}}]
\label{def-LF-annulus}
Let $(h,c)$ be sampled from $\P_\tau\times dc$. Then denote the law of $\phi=h+c$ by $\LF_\tau$ (which is an infinite measure on $H^{-1}(\mathcal{C}_\tau)$).
\end{definition}

For a sample $\phi$ from $\LF_\tau$,
we can similarly define its boundary GMC measures $\nu^1_\phi, \nu_\phi^2$ on the two boundaries $\partial_1\mathcal{C}_\tau=\{0\}\times[0,1]\mathord\sim$ and $\partial_2\mathcal{C}_\tau=\{1\}\times[0,1]\mathord\sim$, respectively. We also let $\{\LF_\tau(\ell_1,\ell_2)\}_{\ell_1,\ell_2>0}$ be the disintegration of $\LF_\tau$ over $\nu_\phi^1(\partial\cC_1)$ and $\nu_\phi^2(\partial\cC_2)$, i.e.~$\LF_\tau=\iint_{\R_+^2}\LF_\tau(\ell_1,\ell_2)d\ell_1d\ell_2$. The following result from~\cite{ARS-Annulus} gives the exact solvability of $|\LF_\tau(\ell_1,\ell_2)|$, which is based on~\cite{wu-annulus}.

\begin{proposition}[{\cite[Equation (3.6)]{ARS-Annulus}}]
 \label{prop-LF-integrability}
For $\tau>0$ and $y\in(-1,\frac{4}{\gamma^2})$, we have
$$
\iint_{\R_+^2}\ell_1e^{-\ell_1}\ell_2^{y}|\LF_\tau(\ell_1,\ell_2)|d\ell_1d\ell_2=\frac{\pi\gamma y\Gamma(1+y)}{2\sin(\frac{\gamma^2}{4}\pi y)}e^{\frac{\pi}{4}\gamma^2\tau y^2}.
$$
\end{proposition}

For two domains $D,\wt{D}\subset\C$ and a conformal map $g:D\to\wt{D}$, when $h$ is a distribution on $D$, we define $g\bullet h:=h\circ g^{-1}+Q\log|(g^{-1})'|$ (which is a distribution on $\wt D$). Let $\Omega$ be the space of simple loops in $\hH$ surrounding $i$, and $\Conf(\hH,i)$ be the group of conformal automorphisms of $\hH$ that fix $i$. For $(\phi,\eta,g,\theta)\in H^{-1}(\hH)\times\Omega\times \Conf(\hH,i)\times[0,1]$, define a measurable map $F$ by
$$
F(\phi,\eta,g,\theta):=\left((g\circ \psi_\eta^{-1})\bullet \phi|_{D(\eta)}, f_{\eta}^\theta\bullet \phi|_{\hH\backslash\overline{D(\eta)}},\Mod(\hH\backslash\overline{D(\eta)})\right).
$$
Here $\psi_\eta:\hH\to D(\eta)$ is the conformal map such that $\psi_\eta(i)=i$ and $\psi_\eta'(i)>0$, and
$f_{\eta}^\theta:\hH\backslash\overline{D(\eta)}\to\mathcal{C}_\tau$ is the conformal map such that $f_{\eta}^\theta(0)=\theta i$ with $\tau=\Mod(\hH\backslash\overline{D(\eta)})$.

Let $\Haar_{(\hH,i)}$ be the Haar measure on $\Conf(\hH,i)$ such that $|\Haar_{(\hH,i)}|=1$, and let $\Unif_{[0,1]}$ be the uniform measure on $[0,1]$.
The following proposition, which is essentially from~\cite{ARS-Annulus}, describes the law of $\LF_\hH^{(\gamma,i)}$ when cut by a simple loop sampled from $\sm$.

\begin{proposition}\label{prop-sle-weld}
There is a constant $C>0$ such that for any $\ell_1>0$,
$$
F_*\left(\LF_{\hH}^{(\gamma,i)}(\ell_1)\times\sm\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)
=C\int_0^\infty \left(\int_0^\infty \ell_2\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi_1)\times\LF_\tau(\ell_2,\ell_1)(d\phi_2)d\ell_2\right)
\eta(2i\tau)d\tau.
$$
Here $F_*$ stands for the pushforward of measures, and we view the right side as a measure on $(\phi_1,\phi_2,\tau)$.
\end{proposition}
\begin{proof}
By~\cite{ARS-Annulus}, the $\SLE_{8/3}$ loop cut the Brownian disk into an independent pair of a (smaller) Brownian disk and a Brownian annulus; see~\cite[Proposition 4.5]{cfsx} for the precise statement and proof. The result then follows from that the uniform embedding of the Brownian disk on $\hH$ gives the Liouville field on $\hH$~\cite[Theorem 3.4]{ARS-FZZ}.
\end{proof}

For $\alpha\in\R$, let $\Delta_\alpha=\frac{\alpha}{2}(Q-\frac{\alpha}{2})$. Define the measure $\sm^\alpha$ via $\frac{d\sm^\alpha}{d\sm}(\eta)=\psi_\eta'(i)^{2\Delta_\alpha-2}$, where $\psi_\eta:\hH\to D(\eta)$ is the conformal map as above. The following proposition is obtained from Proposition~\ref{prop-sle-weld} by a standard reweighting argument. Such an argument was first developed in~\cite{AHS-SLE-integrability}, and appeared in many recent papers on the integrability of SLE/CLE, see e.g.~\cite{ARS-FZZ,ACSW24b,ARS-Annulus,nolin2024backboneexponenttwodimensionalpercolation}.

\begin{proposition}
 \label{prop-sle-reweight}
There is a constant $C>0$ such that for any $\ell_1>0$ and $\alpha\in\R$,
$$
F_*\left(\LF_{\hH}^{(\alpha,i)}(\ell_1)\times\sm^\alpha\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)
=C\int_0^\infty \left(\int_0^\infty \ell_2\LF_{\hH}^{(\alpha,i)}(\ell_2)(d\phi_1)\times\LF_\tau(\ell_2,\ell_1)(d\phi_2)d\ell_2\right)
\eta(2i\tau)d\tau.
$$
As in Proposition~\ref{prop-sle-weld}, we view the right side above as a measure on $(\phi_1,\phi_2,\tau)$.
\end{proposition}

\begin{proof}
For $\phi\in H^{-1}(\hH)$, $\eta\in\Omega$ and $g\in\Conf(\hH,i)$, let $\phi_1=(g\circ \psi_\eta^{-1})\bullet \phi|_{D(\eta)}$. Recall that for $\varepsilon>0$, $(\phi_1)_\varepsilon(i)$ is the average of $\phi_1$ over $\partial B_\varepsilon(i)$.
By~\cite[Lemma 4.8]{ARS-FZZ},
as $\varepsilon\to0$, we have
$$
\begin{aligned}
&F_*\left(\varepsilon^{\frac{1}{2}(\alpha^2-\gamma^2)}e^{(\alpha-\gamma)(\phi_1)_\varepsilon(i)}\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi)\times\sm(d\eta)\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)\\
=~&F_*\left(\varepsilon^{\frac{1}{2}(\alpha^2-\gamma^2)}e^{(\alpha-\gamma)(\phi\circ\psi_\eta\circ g^{-1})_\varepsilon(i)}\LF_{\hH}^{(\gamma,i)}(\ell_2)(d\phi)\times|\psi_\eta'(i)|^{(\alpha-\gamma)Q}\sm(d\eta)\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right)\\
\to~& F_*\left(\LF_{\hH}^{(\alpha,i)}(\ell_2)\times\sm^\alpha\times\Haar_{(\hH,i)}\times\Unif_{[0,1]}\right).
\end{aligned}

\label{section-intro}
\subsection{Overview and the main result}

There is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\ge 1$, the Brownian intersection exponent $\zeta_n=\frac{4n^2-1}{12}$~\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\cite{lsw-restriction}: both of the \emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\kappa=\frac{8}{3}$.

Note that in the critical percolation, one can also define the \emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.

To be precise, let $(B_t)_{t\ge0}$ be a planar Brownian motion starting from $0$. Denote $\mathbbm{S}^1:=\{z\in\mathbbm{C}:|z|=1\}$ and $\mathbbm{D}:=\{z\in\mathbbm{C}:|z|<1\}$ to be the unit circle and the unit disk, respectively. Let $\tau_\mathbbm{D}$ be the first hitting time of $\mathbbm{S}^1$ for $(B_t)_{t\ge0}$.
For each $\varepsilon\in(0,\frac{1}{2})$, consider the event
\begin{equation}\label{eq:def-backbone}
\Bac_\varepsilon=\{\exists\ \text{two disjoint subpaths on the trajectory } B[0,\tau_{\mathbbm{D}}] \text{ joining } \varepsilon\mathbbm{S}^1 \text{ and } \frac{1}{2}\mathbbm{S}^1\}.
\end{equation}
Namely, $\Bac_\varepsilon$ happens if there are two continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbbm{C}$ such that $\gamma^1[0,1]\cap\gamma^2[0,1]=\emptyset$, $\gamma^i[0,1]\subset B[0,\tau_{\mathbbm{D}}]$, $\gamma^i(0)\in\varepsilon\mathbbm{S}^1$, and $\gamma^i(1)\in\frac{1}{2}\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\Bac_\varepsilon$ the \emph{backbone event} for the Brownian trajectory $B[0,\tau_\mathbbm{D}]$.

The main result in this paper is the following iterated logarithmic decay for the probability $\mathbbm{P}[\Bac_\varepsilon]$.

The main result in this paper is the following iterated logarithmic decay for the probability $\P[\Bac_\varepsilon]$.

Let $f$ be the conformal map from $D(\ell_1)$ to $\D$ with $f(0)=0$, $f'(0)>0$ and let $\rho=f(\wt\ell_1)$. Note that $\CR(\rho,0)=\frac{\CR(\wt\ell_1,0)}{\CR(\ell_1,0)}$.
The main result of this section is the following exact law of $\CR(\rho,0)$, which is crucial to the final proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.
\begin{proposition}
\label{prop-cr-rho}
For $\lambda\ge0$, we have
\begin{equation}\label{eq:cr-rho}
\E[\CR(\rho,0)^\lambda]=\frac{\sinh(\frac{\pi}{2}\sqrt{12\lambda-1})}{\sqrt{12\lambda-1}}-\frac{2\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{12\lambda-1})}{\sqrt{12\lambda-1}\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}\right)}.
\end{equation}
When $\lambda\in[0,\frac{1}{12})$, the right side of~\eqref{eq:cr-rho} is defined by analytic continuation; see the end of Section~\ref{section-intro}.
\end{proposition}

Let $\nu_\D$ be the law of the loop chosen from the counting measure over $\{\ell_i\}_{i\ge1}$; namely, $\nu_\D$ is such that $\int F(\eta)\nu_\D(d\eta)=\P\left[\sum_{i\ge1}F(\ell_i)\right]$ for any bounded measurable function $F$. The following result, based on~\cite{cfsx} and Theorem~\ref{thm-cr-mod-relation}, gives the explicit law of the conformal radius under $\nu_\D$.
\begin{lemma}\label{lem-CR-BM-counting-ell}
There is a constant $C>0$ such that for $\lambda\ge0$, we have
\begin{equation}
 \label{eq-CR-BM-counting-ell}
 \int\CR(\eta,0)^\lambda\nu_{\D}(d\eta)=C\frac{\sqrt{12\lambda-1}}{\sinh(\frac{\pi}{3}\pi\sqrt{12\lambda-1})}\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{\pi}{12}\sqrt{12\lambda-1}\right)}\right).
 \end{equation}
When $\lambda\in(0,\frac{1}{12})$, the right side of~\eqref{eq-CR-BM-counting-ell} is defined via analytic continuation as before.
\end{lemma}
\begin{proof}
By combining~\cite[Lemma 5.3]{cfsx} with~\cite[Theorem 1.3]{cfsx}, we have $\frac{d\nu_{\D}}{d\SLE^{\lp}_{8/3,\D}}(\eta)=\frac{C_1}{\Mod(\eta,\S^1)}$ for some constant $C_1>0$.
Then by Theorem~\ref{thm-cr-mod-relation}, we find
\begin{equation}\label{eq:cr-integral}
\int\CR(\eta,0)^\lambda\nu_{\D}(d\eta)=C
 \frac{\sqrt{12\lambda-1}}{\sinh(\frac{\pi}{3}\sqrt{12\lambda-1})}\int_0^\infty e^{-(2\lambda-\frac{1}{6})\pi\tau}\frac{\eta(2i\tau)}{\tau}d\tau.
\end{equation}
for some constant $C>0$.
Note that for $a>-\frac{\pi}{6}$, we have $\int_0^\infty e^{-a\tau}\eta(2i\tau)d\tau=\sqrt{\frac{\pi}{2a}}\frac{\sinh\left(\sqrt{\frac{2}{3}\pi a}\right)}{\cosh\left(\sqrt{\frac{3}{2}\pi a}\right)}$ (see e.g.~\cite[Equation (A.4)]{ARS-Annulus}) and
 $$
 \frac{d}{da}\left[-\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}{(2-\sqrt{3})^2+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}\right)\right]=\sqrt{\frac{\pi}{2a}}\frac{\sinh\left(\sqrt{\frac{2}{3}\pi a}\right)}{\cosh\left(\sqrt{\frac{3}{2}\pi a}\right)}.
 $$
Therefore, we obtain
 $$
 \int_0^\infty e^{-a\tau}\frac{\eta(2i\tau)}{\tau}d\tau=\log\left(\frac{(2+\sqrt{3})^{2}+\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}{1+(2+\sqrt{3})^{2}\tanh^{2}\left(\frac{1}{2}\sqrt{\frac{\pi}{6}a}\right)}\right).
 $$
for $a>-\frac{\pi}{6}$. Combined with the right side of~\eqref{eq:cr-integral}, we conclude.
\end{proof}

The upper bound of $\P[\mathsf{Bac}_\varepsilon]$ then relies on the following iterative inequality.
\begin{proposition}
 \label{prop-upper-bound-backbone}
 Let $q_0=\P[\wt\ell_1\cap\A_{1/2}\neq\emptyset]\in(0,1)$. Then
 there is a constant $C_0>0$ such that for any $\varepsilon\in(0,10^{-4})$,
 $$
 \P[\Bac_\varepsilon]\le \frac{C_0}{\log|\log\varepsilon|}+q_0\P[\Bac_{5\sqrt{\varepsilon}}].
 $$
\end{proposition}
\begin{proof}
Note that
 \begin{align}
 \P[\Bac_\varepsilon]&\le\P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\Bac_\varepsilon]\nonumber\\
 &\le \P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\Bac_\varepsilon]\nonumber\\
 &= \P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset, \Bac_\varepsilon]\label{eq:upper-1}
 \end{align}
Here the last line follows from that $(\wt\ell_1\cap(5\sqrt{\varepsilon}\D\cup\A_{1/2})=\emptyset)\cap\Bac_\varepsilon=\emptyset$.
By Lemma~\ref{lem-CR-BM-ell1}, Corollary~\ref{cor-estimate-cr-rho} and using Koebe's 1/4 theorem, there exist some positive constants $C,C'$ and $C_0$ such that
\begin{align}
\P[\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]+\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D\neq\emptyset]&\le\P[\CR(\ell_1,0)<20\sqrt{\varepsilon}]+\P[\CR(\wt\ell_1,0)<20\sqrt{\varepsilon}]\nonumber\\
 & \le C\varepsilon^{\frac{1}{8}}+C'\frac{1}{\log|\log\varepsilon|}\le \frac{C_0}{\log|\log\varepsilon|}.\label{eq:upper-2}
\end{align}

Let $q_0:=\P[\wt\ell_1\cap\A_{1/2}\neq\emptyset]\in(0,1)$. Then
\begin{equation}\label{eq:upper-3}
\P[\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset,\Bac_\varepsilon]\le q_0\P[\Bac_\varepsilon|\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset].
\end{equation}
Let $g:D(\wt\ell_1)\to\D$ be the conformal map such that $g(0)=0,g(B_{s_1})=1$.
By Lemma~\ref{lem-independent-decomposition-BM}, we see that $(g(B_t))_{t\in[0,s_1]}$ is independent of $B[s_1,\tau_\D]$, and has the same law (up to a time change) as $B[0,\tau_{\D}]$ conditioned on $B_{\tau_\D}=1$. Hence, if we denote $\wt\P$ to be the law of $(g(B_t))_{t\in[0,s_1]}$ conditioned on the event $\{\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset\}\cap\{\wt\ell_1\cap\A_{1/2}\neq\emptyset\}$, then $\wt\P$ is still the same as (up to a time change) the law of $B[0,\tau_{\D}]$ conditioned on $B_{\tau_\D}=1$.
By Lemma~\ref{lem-variant-Koebe-1/4},
on the event $\{\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset\}\cap\{\wt\ell_1\cap\A_{1/2}\neq\emptyset\}$,
we have $\overline{g(\varepsilon\D)}\subset5\sqrt{\varepsilon}\D$ and
$g(\frac{1}{2}\D\cap D(\wt\ell_1))\supset \frac{1}{2}\D$. Then we find
\begin{equation}\label{eq:upper-4}
\P[\Bac_\varepsilon|\wt\ell_1\cap5\sqrt{\varepsilon}\D=\emptyset,\wt\ell_1\cap\A_{1/2}\neq\emptyset]\le \wt\P[\Bac_{5\sqrt{\varepsilon}}]=\P[\Bac_{5\sqrt{\varepsilon}}],
\end{equation}
where the last equality follows from the rotational invariance of $B[0,\tau_\D]$. The result then follows from combining~\eqref{eq:upper-1},~\eqref{eq:upper-2},~\eqref{eq:upper-3} and~\eqref{eq:upper-4}.
\end{proof}

Now we prove the upper bound for $\P[\Bac_\varepsilon]$, finishing the proof of Theorem~\ref{thm:backbone}.
\begin{proof}[Proof of Theorem~\ref{thm:backbone}, the upper bound]
Let $\varepsilon\in(0,10^{-4})$ and $F(\varepsilon):=\P[\Bac_\varepsilon]$. Note that the function $x\mapsto\frac{1}{\log|\log x|}$ is increasing on $(0,10^{-4})$ and there is a constant $C>0$ such that $\frac{1}{\log|\log(25x)|}\le\frac{C}{\log|\log x|}$ for any $x\in(0,10^{-4})$. Then for each $n\in\mathbbm{Z}_+$ with $25\varepsilon^{\frac{1}{2^n}}<10^{-4}$, by applying Proposition~\ref{prop-upper-bound-backbone} for $n$ times, we find
\begin{align}
 F(\varepsilon)
 &\le C_0\sum\limits_{k=0}^{n-1}\frac{q_0^k}{\log|\log(5^{2-\frac{1}{2^{k-1}}}\varepsilon^{\frac{1}{2^k}})|}+q_0^nF(5^{2-\frac{1}{2^{n-1}}}\varepsilon^{\frac{1}{2^n}})\nonumber\\
 &\le \frac{C_0}{1-q_0}\frac{1}{\log|\log(25\varepsilon^{\frac{1}{2^n}})|}+q_0^n\le\frac{C_0C}{1-q_0}\frac{1}{\log|\log(\varepsilon^{\frac{1}{2^n}})|}+q_0^n\nonumber\\
 &=\frac{C_0C}{1-q_0}\frac{1}{\log|\log(\varepsilon)|-n\log2}+q_0^n.\label{eq-proof-upper-bound-backbone}
\end{align}

\begin{remark}
The same result also holds when $\frac{1}{2}\mathbbm{S}^1$ in~\eqref{eq:def-backbone} is changed to any fixed $r\mathbbm{S}^1$ with $r\in(0,1)$ (and for sufficiently small $\varepsilon$), except that the corresponding constants in~\eqref{eq:bb-proba} will depend on $r$.
\end{remark}

Note that the backbone event $\Bac_\varepsilon$ is closely related to the spatial distribution of the \emph{cut points} of $B[0,\tau_\mathbbm{D}]$. Indeed, our proof of Theorem~\ref{thm:backbone} is based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}. We mention that~\cite{cfsx} heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG). It would be quite interesting to see if there is a derivation of Theorem~\ref{thm:backbone} without relying on LQG.

Theorem~\ref{thm:backbone} is related to a certain kind of special points of Brownian motion. Namely, let
\[
\mathcal{B}:=\{B_s:s\in[0,\tau_\mathbbm{D}) \text{ and there exists } \varepsilon>0 \text{ such that } (B_{s+u})_{0\le u\le\varepsilon} \text{ does not have a cut point}\}.
\]
Then Theorem~\ref{thm:backbone} indicates that $\mathcal{B}$ is non-empty and has Hausdorff dimension 2. Furthermore, it also suggests that the Hausdorff measure of $\mathcal{B}$ with the gauge $r\mapsto r^2\log\frac{1}{r}\log\log\frac{1}{r}\log\log\log\frac{1}{r}$~would exist and be non-trivial (since the Hausdorff gauge of $B[0,\tau_\mathbbm{D}]$ itself is $r\mapsto r^2\log\frac{1}{r}\log\log\log\frac{1}{r}$, see e.g.~\cite{LG85}). In particular, due to that planar Brownian motion a.s.~has no double cut points~\cite{BL}, we know that $\mathcal{B}$ contains the set of double points of $(B_t)_{0\le t\le\tau_\mathbbm{D}}$, which has the Hausdorff gauge $r\mapsto r^2(\log\frac{1}{r}\log\log\log\frac{1}{r})^2$~\cite{LG87}.

\subsection{Outlook and discussions}
\label{section-discussion}

Here we give several remarks and related questions before going into the proof.

\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}

\item One can similarly consider the probability that there exist $2n$ disjoint subpaths on $B[0,\tau_\mathbbm{D}]$ joining $\varepsilon\mathbbm{S}^1$ and $\frac{1}{2}\mathbbm{S}^1$ with $n\ge2$. Note that for $n\ge2$, such $2n$-arm probability is not straightforwardly related to the cut points of $B[0,\tau_\mathbbm{D}]$. Instead, one need to consider the $(2n-1)$-tuples of the local cut points such that removing these $(2n-1)$ cut points from the trajectory $B[0,\tau_\mathbbm{D}]$, $0$ and $B_{\tau_\mathbbm{D}}$ are not in the same connected component of the remaining set. However, analyzing such tuples of local cut points (e.g.~the counterpart of the layer structure in Section~\ref{section-backbone-1}) becomes much more complicated, and it seems difficult to solve them explicitly. We mention that similar difficulty also appears in deriving the monochromatic $k$-arm exponents of critical planar percolation for $k\ge3$, see~\cite[Remark 2.3]{nolin2024backboneexponenttwodimensionalpercolation}.

\item
There is also a natural half-plane variant of our setup. Namely, let $(e_t)_{t\ge0}$ be the Brownian excursion on the upper half plane $\mathbbm{H}$ from $0$ to $\infty$, and define $R\mathbbm{S}^+:=R\mathbbm{S}^1\cap\mathbbm{H}$ for $R>0$. Then for $n\ge1$, consider the asymptotic probability that
there exists $(2n+1)$ disjoint subpaths on the trajectory of $(e_t)$ joining $\mathbbm{S}^+$ and $R\mathbbm{S}^+$ as $R\to\infty$.
Note that the case $n=1$ can be similarly related to the cut points of $(e_t)$, which has been proven in~\cite[Theorem 4]{brownian-beads}.

\item In our forthcoming work~\cite{cx}, we will extend the result of this paper to the Brownian loop soup cases. Let $(B_t)_{0\le t\le\tau_\mathbbm{D}}$ be as before, and let $\mathcal{L}$ be an independent Brownian loop soup on $\mathbbm{D}$ with intensity $\frac{c}{2}$ for $c\in(0,1]$.
For a subset $A\subset\ol\mathbbm{D}$, denote $\mathcal{C}(A)$ to be the union of $A$ and all loop-soup clusters in $\mathcal{L}$ intersecting with $A$,
and let $H:=\mathcal{C}(B[0,\tau_\mathbbm{D}])$.
However, unlike the Brownian motion case,
there is a positive probability that $0$ itself can have two disjoint subpaths (except on $0$) on $H$
connected to $\frac{1}{2}\mathbbm{S}^1$. Moreover, in~\cite{cx} we will show the following
\begin{theorem}
We say $t\in(0,\tau_\mathbbm{D}]$ is a cut time of $H$ if $\mathcal{C}(B[0,t])\cap \mathcal{C}(B[t,\tau_\mathbbm{D}])=B_t$. Let $t_1$ be the largest cut time of $H$ such that the boundary of $\mathcal{C}(B[0,t_1])$ is a simple loop, and denote $E$ to be the event that $t_1$ is the smallest cut time of $H$.
Then for $c\in(0,1)$,
\begin{equation}\label{eq:p_e}
\mathbbm{P}[E]=2^{\frac{c+1}{2}}\sqrt{\frac{6}{1-c}}\left(\int_0^\infty\tau^{-1-\frac{c}{2}}e^{\frac{1-c}{6}\pi\tau}\eta(2i\tau)^{1-c}d\tau\right)^{-1}
\end{equation}
where $\eta(2i\tau):=e^{-\frac{\pi}{6}\tau}\prod_{n=1}^{\infty}\left(1-e^{-4\pi n\tau}\right)$ is the Dedekind eta function.
\end{theorem}
Note that the right side of~\eqref{eq:p_e} tends to $1$ as $c\uparrow1$, and tends to $0$ as $c\downarrow0$.

\end{itemize}

\medskip
\noindent\textbf{Organization of the paper.} In Section~\ref{section-SLE loop}, we review the explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops. Such relation was implicitly established in~\cite{ARS-Annulus}, and we will provide its proof in Appendix~\ref{appendix} for completeness. Then in Section~\ref{section-backbone-1}, we define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and solve it explicitly, based on the results of~\cite{cfsx} and Section~\ref{section-SLE loop}. Finally, we finish the proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.

\medskip
\noindent\textbf{Basic notations.}
For two compact sets $A,B\subset\mathbbm{C}$, let $\mathrm{dist}(A,B):=\inf\{|a-b|:a\in A,b\in B\}$.
For a simple loop $\ell\subset\mathbbm{C}$, let $D(\ell)$ be the bounded connected component of $\mathbbm{C}\backslash\ell$. For a simple loop $\ell\subset\mathbbm{C}$ with $0\in D(\ell)$, we denote the conformal radius of $D(\ell)$ seen from 0 by $\mathrm{CR}(\ell,0)$, i.e. if $f:\mathbbm{D}\to D(\ell)$ is a conformal map that fixes the origin, then $\mathrm{CR}(\ell,0)=|f'(0)|$.

For $0<r<1$, let $\A_r$ be the standard annulus $\{z\in\mathbbm{C}: r<|z|<1\}$. For each annular domain $A\subset\mathbbm{C}$,
there exists a unique $\tau>0$ such that $A$ and $\A_{e^{-2\pi\tau}}$ are conformally equivalent. We call $\tau$ the \textit{modulus} of $A$ and write $\mathrm{Mod}(A):=\tau$. For two simple loops $\eta_1,\eta_2$ such that $\overline{D(\eta_1)}\subset D(\eta_2)$, we also write the modulus of the annular domain $D(\eta_2)\setminus\overline{D(\eta_1)}$ as $\mathrm{Mod}(\eta_1,\eta_2)$ for simplicity.

We will frequently deal with elementary functions such as $\sqrt{x},\sinh(\sqrt{x}),\cosh(\sqrt{x})$, and $\tanh(\sqrt{x})$. In the following, we view them as functions defined on $\mathbbm{R}$ by taking $i\sqrt{|x|},i\sin(\sqrt{|x|}),\cos(\sqrt{|x|})$, and $i\tan(\sqrt{|x|})$ for $x<0$, respectively.

\medskip
\noindent\textbf{Acknowledgment.}
We are grateful to Xin Sun for many helpful discussions and suggestions on the early draft of this paper.
G.C. and Z.X.\ were partially supported by National Key R\&D Program of China (No.\ 2023YFA1010700). G.C. was partially supported by National Key R\&D Program of China (No. 2021YFA1002700).

Our proof of Theorem~\ref{thm:backbone} is described as follows: it is “based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Section~\ref{section-SLE loop} reviews an “explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops”; Section~\ref{section-backbone-1} then defines the layer structure and “solve[s] it explicitly, based on the results of~\cite{cfsx} and Section~\ref{section-SLE loop}”; finally “we finish the proof of Theorem~\ref{thm:backbone} in Section~\ref{section-backbone-2}.”

Our proof of the main theorem is described as follows: it is “based on our recent paper with Fu and Sun~\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\tau_\mathbbm{D}]$ and then solve it explicitly; see Section~\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Next we review an “explicit relation between the laws of conformal radii and moduli for $\SLE_{8/3}$-type loops”; we then define the layer structure and “solve[s] it explicitly, based on the results of~\cite{cfsx} and the preceding review”; finally “we finish the proof of the main theorem later.”

Existential–Universal
Inequality or Bound

Question:
Let $(B_t)_{t\ge 0}$ be planar Brownian motion started from $0$, let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, and let $\tau_{\mathbb D}$ be the first time that $B_t$ hits the unit circle $\mathbb S^1=\{z\in\mathbb C:|z|=1\}$. For $\varepsilon\in(0,\tfrac12)$, define the backbone event
\[
\mathrm{Bac}_\varepsilon=
\Big\{\text{there exist two disjoint subpaths of }B[0,\tau_{\mathbb D}]\text{ joining }\varepsilon\mathbb S^1\text{ and }\tfrac12\mathbb S^1\Big\}.
\]
Equivalently, there exist continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbb C$ with disjoint images, each contained in the Brownian trajectory $B[0,\tau_{\mathbb D}]$, such that $\gamma^i(0)\in \varepsilon\mathbb S^1$ and $\gamma^i(1)\in \tfrac12\mathbb S^1$ for $i=1,2$. Which quantitative estimate holds for $\mathbb P[\mathrm{Bac}_\varepsilon]$ for small $\varepsilon$?

There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]

Choices:
B.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
C_1\,\frac{1}{|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le C_2\,\frac{1}{|\log \varepsilon|}.
\]
C.
There exists a constant $C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
D.
There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac12)$,
\[
\frac{C_1}{\log|\log \varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log \varepsilon|}.
\]
E.
For every $\delta>0$, there exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,
\[
\frac{C_1}{(\log|\log \varepsilon|)^{1+\delta}}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{(\log|\log \varepsilon|)^{1-\delta}}.
\]


Question: Let $(B_t)_{t\ge 0}$ be planar Brownian motion started at $0$, let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, let $\mathbb S^1=\{z\in\mathbb C:|z|=1\}$, and let $\tau_{\mathbb D}$ be the first time that $B_t$ hits $\mathbb S^1$. For $\varepsilon\in(0,\tfrac12)$, define the backbone event $\mathrm{Bac}_\varepsilon$ to be the event that there exist two disjoint subpaths of the Brownian trajectory $B[0,\tau_{\mathbb D}]$ joining $\varepsilon\mathbb S^1$ to $\tfrac12\mathbb S^1$; equivalently, there exist continuous curves $\gamma^1,\gamma^2:[0,1]\to\mathbb C$ with disjoint images, each image contained in $B[0,\tau_{\mathbb D}]$, such that $\gamma^i(0)\in \varepsilon\mathbb S^1$ and $\gamma^i(1)\in \tfrac12\mathbb S^1$ for $i=1,2$. Which explicit statement holds for the probability of $\mathrm{Bac}_\varepsilon$?

There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,\[\frac{C_1}{\log|\log\varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log\varepsilon|}.\]

Choices:
B. There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,\[\frac{C_1}{|\log\varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{|\log\varepsilon|}.\]
C. There exists a constant $C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,\[\mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log\varepsilon|}.\]
D. There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{2})$,\[\frac{C_1}{\log|\log\varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log\varepsilon|}.\]
E. There exist constants $C_1,C_2>0$ such that for every $\varepsilon\in(0,\tfrac{1}{10})$,\[\frac{C_1}{\log|\log\varepsilon|}\le \mathbb P[\mathrm{Bac}_\varepsilon]\le \frac{C_2}{\log|\log\varepsilon|^2}.\]

iterated-log decay replaced by single-log decay

dropped the lower bound while keeping the proved upper bound scale

validity range in $\varepsilon$ extended from $(0,\tfrac{1}{10})$ to $(0,\tfrac{1}{2})$

asymmetric strengthening of the upper bound to a smaller order than the theorem gives