[Rank-decomposition and rankwidth \cite{Oum_Seymour_2006, Voigt}] Given a graph $G=(V, E)$, a \emph{rank-decomposition} of $G$ is a pair $(T,\gamma)$ where $T$ is a subcubic tree (i.e., every internal node has degree at most $3$) and $\gamma\colon V(G)\to\operatorname{Leaf}(T)$ is a bijection. Observe that
deleting any edge $e\in E(T)$ splits the set of leaves into parts $(X_e,Y_e)$, thereby inducing a cut in $G$.
Then, the rankwidth of $G$ is given by
\begin{align}\mathrm{rankwidth}(G) \coloneqq \min_{(T,\gamma)} \max_{e\in E(T)}\operatorname{cutrank}_G(X_e,Y_e).\end{align}

The relationship between combinatorial width parameters and connectivity properties is a cornerstone of structural and algorithmic graph theory \cite{BODLAENDER19981, SEYMOUR199322, ROBERTSON1994323, harvey2016parameterstiedtreewidth, BOTTCHER20101217}.
For tree-width, the celebrated Grid-Minor Theorem of Chekuri and Chuzhoy \cite{10.1145/2820609} and its recent developments \cite{doi:10.1137/1.9781611977912.48, Geelen_2023} characterizes large width via the presence of an $r\times r$ grid minor, while branch-width \cite{fomin2021fastfptapproximationbranchwidth,https://doi.org/10.1002/net.20050} admits a well-linked-set characterization due to Oum and Seymour \cite{Oum_Seymour_2006}. While such relationships are well-understood for treewidth and branchwidth, a corresponding characterization for rankwidth and its sibling, clique-width, remains a gap in the literature: indeed, no well-linkedness or expander-style certificate has been pinned down. Here, we provide a proof of existence of such a certificate by establishing a linear dependence between rankwidth and another parameter, the minimum balanced cutrank. We begin by defining the minimum balanced cutrank. \\

A \emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\subseteq V(G)$ such that $X\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \emph{cutrank} of the separation, $\operatorname{cutrank}(X,Y)$, is the $\mathbb{F}_2$-rank of its $|X|\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \emph{balanced} if $\tfrac13|V(G)|\le|X|$ and $|Y|\le\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \emph{minimum balanced cutrank}, $\operatorname{min-bal-cutrank}(G)$, by
\begin{align}\operatorname{min-bal-cutrank}(G) = \min\mathopen{}\mathclose\bgroup\originalleft\{\operatorname{cutrank}(X,Y)\text{ }\Big| \text{ }X\sqcup Y = V(G)\text{ with }|X|, |Y| \leq \frac{2n}{3}\aftergroup\egroup\originalright\}. \end{align}

Informally, a graph whose minimum balanced cutrank is large should have no bisection with low rank, and therefore behaves like an \emph{expander graph} when measured in $\mathbb{F}_2$ rank. A vertex set is said to be an expander if it has many external neighbors. We call a graph bounded if every sufficiently large subgraph contains a subset which is not expanding. There is a long line of literature on expander graphs for their ubiquitous use in pseudorandomness \cite{Alon1986Eigenvalues,anand2025pseudorandomnessstickyrandomwalk,golowich_et_al:LIPIcs.CCC.2022.27,anand2025pseudorandomnessexpanderrandomwalks}, unique games \cite{bafna2021highdimensionalexpanderseigenstripping}, log-space computation \cite{Reingold2008}, the PCP theorem \cite{10.1145/1236457.1236459}, fault-tolerant computation \cite{AjtaiAlonBruckEtAl1992,chaudhari2025peer}, and a number of other applications. We refer the interested reader to the (incredible) surveys of Lubotzky \cite{lubotzky2011expandergraphspureapplied} and Hoory, Linial, and Wigderson \cite{Hoory2006}. However, the combinatorial notion of expansion is only meaningful for sparse graphs, and it is desirable in a number of applications to have an analog of expansion for dense graphs \cite{alon2003densegraphsantimagic, alon1992almost, doi:10.1137/18M122039X, 10.1016/0012-365X(95)00242-O, Alon_Nachmias_Shalev_2022, alon1998packing, srivastava2017alonboppanatypeboundweighted}. There are a number of concepts that have been studied in the literature previously, such as flip-width \cite{K_hn_2014,toruńczyk2024flipwidthcopsrobberdense} and twin-width \cite{bonnet2022}, but prior to now there was no agreed-upon notion.\\

For instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \emph{rank-expansion} is based on the $\mathbb{F}_2$-rank of the adjacency matrix.
This offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \ref{thm:main} proposes that \emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

A \emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\subseteq V(G)$ such that $X\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \emph{cutrank} of the separation, $\operatorname{cutrank}(X,Y)$, is the $\mathbb{F}_2$-rank of its $|X|\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \emph{balanced} if $\tfrac13|V(G)|\le|X|$ and $|Y|\le\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \emph{minimum balanced cutrank}, $\operatorname{min-bal-cutrank}(G)$, by
\begin{align}\operatorname{min-bal-cutrank}(G) = \min\left\{\operatorname{cutrank}(X,Y)\text{ }\Big| \text{ }X\sqcup Y = V(G)\text{ with }|X|, |Y| \leq \frac{2n}{3}\right\}. \end{align}

For instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \emph{rank-expansion} is based on the $\mathbb{F}_2$-rank of the adjacency matrix.
This offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \ref{thm:main} proposes that \emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.

Motivated by this definition of rankwidth, we now provide a formal definition of \emph{rank-expansion}.

\begin{definition}[Rank-Expansion]\label{def: rank-expansion}
For a graph $G=(V,E)$ where $|V|=n$, the \emph{rank-expansion} is given by
 \begin{align}\operatorname{rank-expansion}(G) \coloneqq \max_{S\subseteq V(G): |S|\leq n/2} \frac{\operatorname{cutrank}(S, V(G)\setminus S)}{|S|}.\end{align}
\end{definition}

\noindent Theorem~\ref{thm:main} parallels the grid-minor theorem of Chekuri and Chuzhoy \cite{doi:10.1137/1.9781611977912.48, Geelen_2023, 10.1145/2820609}: delete vertices instead of contracting edges, and replace edge cuts by $\mathbb{F}_2$-rank cuts. For any graph $G$, this implies that
\begin{align}
\tfrac1{72}\,\mathrm{rankwidth}(G)\;\le\;
\max_{H\subseteq G}\operatorname{min-bal-cutrank(H)}
\;\le\;
\mathrm{rankwidth}(G).
\end{align}

\noindent Here, the right inequality holds because for any graph $H$ and any rank-decomposition of $H$ of width $k$, there exists a balanced cut $(A, B)$ with $|A|,|B|\geq |V(H)|/3$ such that $\operatorname{cutrank}_H(A,B) \leq k$. Therefore, for every induced $H\subseteq G$, the minimum cutrank (across balanced sets $A$ and $B$) is $\operatorname{cutrank}_H(A,B)\le \mathrm{rankwidth}(H)\le \mathrm{rankwidth}(G)$: by properties of tree decompositions, any rank-decomposition of width $k$ contains a balanced cut of rank at most $k$. The left inequality follows from Theorem \ref{thm:main}, since if $\mathrm{rankwidth}(G)\ge 72r$, then
there is an induced $H$ satisfying $\min_{(A,B)}\operatorname{cutrank}_H(A,B)\ge r$, for some balanced partition $(A,B)$ of $V(H)$.\\

First, in Claim \ref{claim:rank-tangle existence}, we adapt the tangle-and-congestion machinery of Dvořák and Norin \cite{Dvorak_Norin_2018} to extract the more strongly-connected side $X$ of a low-rank separation $(X,Y)$ that is maximally large yet \emph{well-behaved}. This well-behaved property formalizes the idea that $X$ is robustly connected to $Y$, meaning no small component of $X$ can be separated from the rest of the graph with a low-rank cut. Next, we observe that the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$ is bounded. This allows us to find a small subset of representative vertices in $X$ that captures all of these connections, effectively compressing $X$ without losing essential rank information. Specifically, we compress $X$ by taking one representative per $Y$-neighborhood class (Claim \ref{claim:findingH}), which preserves $\operatorname{cutrank}(X,Y)$; then, in Lemma \ref{lemma: splitting Y reduction}, we form an induced subgraph $H'$ so that any balanced cut of $H'$ of rank at most $r$ splits $Y$ and yields two lifted separations of $G$ whose cutranks are at most $72r$; combined with the tangle axiom this yields the existence of a strongly-connected side in $V(G)$ that is strictly larger than $X$, thereby contradicting the maximality assumption.\\

\begin{definition}
 Suppose $(X, Y)$ is a partition of $V(G)$. For each $u,v\in Y$, we define equivalence classes, where each class has the relation $u\sim v$ iff $N_G(u)\cap X = N_G(v) \cap X$. The family $\mathcal F=\{\,N_G(y)\cap X : y\in Y\}$ of subsets of $X$ has \emph{VC-dimension} $\operatorname{VC}(\mathcal F)$, the largest $S\subseteq X$ shattered by $\mathcal F$.
Then, Sauer–Shelah's lemma \cite{SAUER1972145,pjm1102968432} bounds $|\mathcal F|$ (the number of $X$-equivalence classes) by $\sum_{i=0}^{d}\binom{|X|}{i}=O(|X|^{d})$ where $d=\operatorname{VC}(\mathcal F)$.
\end{definition}
Let the number of equivalence classes be denoted by $N_Y$. Then, we have that $\log(N_Y) \leq \operatorname{cutrank}(X,Y) \leq {N_Y}$ since the rank cannot exceed the number of distinct column types (equivalence classes), and over $\mathbb{F}_2$, $k$ dimensions can encode at most $2^k$ distinct neighborhood patterns. Furthermore, if $N_Y \leq O(n^{1-\epsilon})$, then a number of separator theorems emerge, due to the works of Feige, Hajiaghayi, and Lee \cite{Feige_Hajiaghayi_Lee_2008}.

\begin{claim}
\label{claim:findingH}
    There exists an induced subgraph $H\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\in X$, there exists at least one vertex $v'\in X\cap V(H) \eqqcolon X^*$ such that $N(v)\cap Y = N(v')\cap Y$.
\end{claim}

\begin{claim}
\label{claim:rank-tangle existence}
    There exists a partition $(X,Y)$ of $V(G)$ such that $\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \in \tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\tau$ with $\mathrm{cutrank}<70r$ (and, subject to this, $\mathrm{cutrank}(X,Y)$ is minimum).
\end{claim}

\begin{lemma}[Splitting $Y$]
    \label{lemma: splitting Y reduction}
    There exists an induced subgraph $H'\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \cap Y \neq \emptyset$ and $B'\cap Y\neq\emptyset$, $\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \eqqcolon V(H') \cap X^*$, then \[\operatorname{cutrank}_{H'}(X^{**}, Y) \leq \operatorname{cutrank}_H(X^*, Y) = \operatorname{cutrank}_G(X,Y) < 70r.\]
\end{lemma}

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

Motivated by this definition of rankwidth, we now provide a formal definition of \emph{rank-expansion}.

\begin{definition}[Rank-Expansion]\label{def: rank-expansion}
For a graph $G=(V,E)$ where $|V|=n$, the \emph{rank-expansion} is given by
    \begin{align}\operatorname{rank-expansion}(G) \coloneqq \max_{S\subseteq V(G): |S|\leq n/2} \frac{\operatorname{cutrank}(S, V(G)\setminus S)}{|S|}.\end{align}
\end{definition}

We show that large rankwidths of a graph $G$ enforce the existence of a large, rank-robust subgraph $H$.

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

\noindent Theorem~\ref{thm:main} parallels the grid-minor theorem of   Chekuri and Chuzhoy \cite{doi:10.1137/1.9781611977912.48, Geelen_2023, 10.1145/2820609}: delete vertices instead of contracting edges, and replace edge cuts by $\mathbb{F}_2$-rank cuts. For any graph $G$, this implies that
\begin{align}
\tfrac1{72}\,\mathrm{rankwidth}(G)\;\le\;
\max_{H\subseteq G}\operatorname{min-bal-cutrank(H)}
\;\le\;
\mathrm{rankwidth}(G).
\end{align}

\noindent Here, the right inequality holds because for any graph $H$ and any rank-decomposition of $H$ of width $k$, there exists a balanced cut $(A, B)$ with $|A|,|B|\geq |V(H)|/3$ such that $\operatorname{cutrank}_H(A,B) \leq k$. Therefore, for every induced $H\subseteq G$, the minimum cutrank (across balanced sets $A$ and $B$) is $\operatorname{cutrank}_H(A,B)\le \mathrm{rankwidth}(H)\le \mathrm{rankwidth}(G)$: by properties of tree decompositions, any rank-decomposition of width $k$ contains a balanced cut of rank at most $k$. The left inequality follows from Theorem \ref{thm:main}, since if $\mathrm{rankwidth}(G)\ge 72r$, then
there is an induced $H$ satisfying $\min_{(A,B)}\operatorname{cutrank}_H(A,B)\ge r$, for some balanced partition $(A,B)$ of $V(H)$.\\

Oum’s vertex-minor theorem states that rankwidth~$k$ forces a $K_t$ minor with $t\approx2^k$ and provides a set of excluded vertex-minors for graphs of bounded rankwidth, which is analogous to the Robertson-Seymour theorem for graph minors \cite{Oum_Seymour_2006}.
In contrast, our result finds an {induced} subgraph such that  {every} balanced separation of the subgraph has rank $\Theta(k)$. Thus, graphs with large width must contain a highly connected induced subgraph with robust cutrank. Dvořák and Norin show that large tree-width forces large minimum balanced \emph{edge}-separators~\cite{Dvorak_Norin_2018}; in our work, we show how to transfer this to vertex-rank cuts. Finally, Oum and Seymour provide approximation algorithms for rankwidth \cite{Oum_Seymour_2006}; in this spirit, Theorem \ref{thm:main} supplies a proof of existence of an alternative certificate that may lead to new parameter-testing routines. \\

Through our result, we suggest rank-expansion as a candidate notion of expansion suitable for dense graphs. Indeed, our result can analogously be viewed as a Cheeger-type inequality for dense graphs \cite{Cheeger1970,AlonMilman1985}, which adds to the growing literature in the field \cite{khetan2018cheegerinequalitiesgraphlimits,jost2024cheegerinequalitiessimplicialcomplexes,10.1145/3564246.3585139,briani2022classcheegerinequalities, 10.1145/2582112.2582118, schild2018schurcomplementcheegerinequality, first2025cheegerinequalitycoboundaryexpansion}. For instance, Cheeger gives that $c_1(1 - \lambda_2) \leq \phi(G) \leq c_2\sqrt{1-\lambda_2}$, where $\phi(G)$ is the conductance of $G$, and $\lambda_2$ is the second largest (in absolute value) eigenvalue of $G$. Conversely, Theorem \ref{thm:main} gives $\frac{1}{72}\operatorname{rankwidth}(G) \leq \rho(G) \leq \operatorname{rankwidth}(G)$, where $\rho(G)$ is the minimum balanced cutrank of any subgraph of $G$, i.e. $\rho(G) = \max_{H\subseteq G} \operatorname{min-bal-cutrank}_H(A,B)$. In this vein, our work is most related to that of \cite{K_hn_2014} which shows a different result that dense regular graphs can be partitioned into robust expanders, which are subgraphs that still have good expansion properties after removing some vertices and edges. In turn, this can be applied to Hamiltonicity problems in dense regular graphs.

\paragraph{Proof overview.} Our proof proceeds by way of contradiction. If a graph has high rankwidth but low min-bal-cutrank, then every part of the graph can be cleanly separated with a low-rank cut. We introduce the tangle in definition \ref{def: rank-tangle of order k}, which points to the `more connected' component of a separation of a graph. We use it to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected. Subject to this, we assume $X$ is maximal and call it the {small side}, following the convention of \cite{Dvorak_Norin_2018}. We then \emph{compress} $X$ by selecting a subset of {representative} vertices, and show that finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality. Our proof continues via three stages.\\

First, in Claim \ref{claim:rank-tangle existence}, we adapt the tangle-and-congestion machinery of Dvořák and Norin \cite{Dvorak_Norin_2018} to extract the more strongly-connected side $X$ of a low-rank separation $(X,Y)$ that is maximally large yet \emph{well-behaved}. This well-behaved property formalizes the idea that $X$ is robustly connected to $Y$, meaning no small component of $X$ can be separated from the rest of the graph with a low-rank cut. Next, we observe that the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$ is bounded. This allows us to find a small subset of representative vertices in $X$ that captures all of these connections, effectively compressing $X$ without losing essential rank information. Specifically, we compress $X$ by taking one representative per $Y$-neighborhood class (Claim \ref{claim:findingH}), which preserves $\operatorname{cutrank}(X,Y)$; then, in Lemma \ref{lemma: splitting Y reduction}, we form an induced subgraph $H'$  so that any balanced cut of $H'$ of rank at most $r$ splits $Y$ and yields two lifted separations of $G$ whose cutranks are at most $72r$; combined with the tangle axiom this yields the existence of a strongly-connected side in $V(G)$ that is strictly larger than $X$, thereby contradicting the maximality assumption.\\

\paragraph{Proof overview.} The proof is "by way of contradiction": assuming "a graph has high rankwidth but low min-bal-cutrank," so that "every part of the graph can be cleanly separated with a low-rank cut." The argument introduces "the tangle in definition \\ref{def: rank-tangle of order k}, which points to the `more connected' component of a separation," and uses it "to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected." With $X$ chosen "maximal" (the "small side"), one then "compress[es] $X$ by selecting a subset of representative vertices," and shows that "finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality."

The proof proceeds in three stages: (1) In Claim~\\ref{claim:rank-tangle existence}, "adapt the tangle-and-congestion machinery of Dvořák and Norin" to extract a low-rank separation $(X,Y)$ where $X$ is "maximally large" and "well-behaved," meaning "$X$ is robustly connected to $Y$" ("no small component of $X$ can be separated from the rest of the graph with a low-rank cut"). (2) Bound "the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$" and thereby find "a small subset of representative vertices in $X$"; concretely, "compress $X$ by taking one representative per $Y$-neighborhood class (Claim~\\ref{claim:findingH}), which preserves $\\operatorname{cutrank}(X,Y)$." (3) In Lemma~\\ref{lemma: splitting Y reduction}, form an induced subgraph $H'$ so that "any balanced cut of $H'$ of rank at most $r$" forces a split of $Y$ and gives "two lifted separations of $G$ whose cutranks are at most $72r$"; "combined with the tangle axiom" this yields "a strongly-connected side" strictly larger than $X$, contradicting maximality, and thus establishing Theorem~\\ref{thm:main}.

\paragraph{Proof overview.} The proof is "by way of contradiction": assuming "a graph has high rankwidth but low min-bal-cutrank," so that "every part of the graph can be cleanly separated with a low-rank cut." The argument introduces the following definition.

\begin{definition}[Rank-tangle of order $k$] \label{def: rank-tangle of order k}
Following Robertson-Seymour \cite{ROBERTSON1991153}, a rank-tangle of order $k$ is a set $\mathcal{T}$ of subsets of $V(G)$ such that \begin{enumerate}\item For every partition $(X,Y)$ of $V(G)$ with $\operatorname{cutrank}_G(X,Y)<k$, exactly one of $X$ or $Y$ is in $\mathcal{T}$ (and is called the ``small side'')\item No three chosen small sides have union $V(G)$; i.e., for any sets $S_1, S_2, S_3 \in \mathcal{T}$, their union $S_1 \cup S_2 \cup S_3 \neq V(G)$.\end{enumerate}
The existence of a rank-tangle of order $k$ witnesses that $\mathrm{rankwidth}(G)\geq k$.
\end{definition}

This tangle points to the `more connected' component of a separation, and we use it to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected. With $X$ chosen "maximal" (the "small side"), one then "compress[es] $X$ by selecting a subset of representative vertices," and shows that "finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality."

The proof proceeds in three stages: (1) We use the following claim to extract a low-rank separation $(X,Y)$ where $X$ is "maximally large" and "well-behaved," meaning "$X$ is robustly connected to $Y$" ("no small component of $X$ can be separated from the rest of the graph with a low-rank cut").

\begin{claim}
\label{claim:rank-tangle existence}
    There exists a partition $(X,Y)$ of $V(G)$ such that $\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \in \tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\tau$ with $\mathrm{cutrank}<70r$ (and, subject to this, $\mathrm{cutrank}(X,Y)$ is minimum).
\end{claim}

(2) We bound "the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$" and thereby find "a small subset of representative vertices in $X$"; concretely, we "compress $X$ by taking one representative per $Y$-neighborhood class" using the following claim, which preserves $\operatorname{cutrank}(X,Y)$.

\begin{claim}
\label{claim:findingH}
    There exists an induced subgraph $H\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\in X$, there exists at least one vertex $v'\in X\cap V(H) \eqqcolon X^*$ such that $N(v)\cap Y = N(v')\cap Y$.
\end{claim}

(3) We then use the following lemma to form an induced subgraph $H'$ so that "any balanced cut of $H'$ of rank at most $r$" forces a split of $Y$ and gives "two lifted separations of $G$ whose cutranks are at most $72r$"; combined with the tangle axiom, this yields a strongly-connected side strictly larger than $X$, contradicting maximality.

\begin{lemma}[Splitting $Y$]
    \label{lemma: splitting Y reduction}
    There exists an induced subgraph $H'\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \cap Y \neq \emptyset$ and $B'\cap Y\neq\emptyset$, $\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \eqqcolon V(H') \cap X^*$, then \[\operatorname{cutrank}_{H'}(X^{**}, Y) \leq \operatorname{cutrank}_H(X^*, Y) = \operatorname{cutrank}_G(X,Y) < 70r.\]
\end{lemma}

This contradiction completes the proof of the following theorem.

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

Classification or Bijection
Inequality or Bound

Question: Let $G=(V,E)$ be a graph. A separation of $G$ is a pair $(X,Y)$ with $X\sqcup Y=V(G)$, and $\operatorname{cutrank}_G(X,Y)$ is the $\mathbb F_2$-rank of the $|X|\times |Y|$ biadjacency matrix between $X$ and $Y$. A rank-decomposition of $G$ is a pair $(T,\gamma)$ where $T$ is a subcubic tree (every internal node has degree at most $3$) and $\gamma:V(G)\to \operatorname{Leaf}(T)$ is a bijection. For an edge $e\in E(T)$, deleting $e$ partitions the leaves of $T$ into two parts, which via $\gamma^{-1}$ induce a partition $(X_e,Y_e)$ of $V(G)$. Which quantitative formula gives the rankwidth of $G$?

$\displaystyle \operatorname{rankwidth}(G)=\min_{(T,\gamma)}\max_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$, where the minimum is over all rank-decompositions $(T,\gamma)$ of $G$.

Choices:
B. $\displaystyle \operatorname{rankwidth}(G)=\max_{(T,\gamma)}\min_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$, where the maximum is over all rank-decompositions $(T,\gamma)$ of $G$.
C. $\displaystyle \operatorname{rankwidth}(G)\le \max_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$ for every rank-decomposition $(T,\gamma)$ of $G$.
D. $\displaystyle \operatorname{rankwidth}(G)=\min_{(T,\gamma)}\sum_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$, where the minimum is over all rank-decompositions $(T,\gamma)$ of $G$.
E. $\displaystyle \operatorname{rankwidth}(G)=\min_{(T,\gamma)}\max_{e\in E(T)} |X_e|\,|Y_e|$, where the minimum is over all rank-decompositions $(T,\gamma)$ of $G$.

min-max order in the defining optimization

global minimum over decompositions dropped, keeping only the upper-bound inequality for each fixed decomposition

width measured by worst edge replaced by total aggregate over edges

cutrank replaced by cut size proxy depending only on part sizes


\label{th:main}
Fix a truncation parameter $m_0\ge1$ and
assume that:
\begin{enumerate}[---]
\item the space dimension is sufficiently large depending on $m_0$, in the sense that
\begin{equation}\label{eq:def de d_0}
d\ge d_0=\left\{\begin{array}{cl}
2,&\text{if $m_0=1$},\\
8,&\text{if $m_0=2$},\\
28m_0+70,&\text{if $m_0\ge3$},
\end{array}\right.\end{equation}
\item the diffusion constant $\kappa$ is sufficiently large, in the sense that $\kappa\ge C_{m_0}\int_{\mathbb R^d}|k|\V(k)\,dk$ for some constant~$C_{m_0}>0$ depending only on $m_0$. In case $m_0=1$, this condition can be dropped.
\end{enumerate}
Then, for any initial condition $\mathfrak G\in C_c^\infty(\mathbb R^d)$, the solution $(G_m^{N,m_0})_{0\le m\le m_0}$ of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci} satisfies that the time-rescaled tagged-particle density
\[(\tau,v)\mapsto G_0^{N,m_0}(N\tau,v)\]
converges strongly in $L^2_\loc(\mathbb R^+;L^2(M\, dv))$ to the solution $G_0$ of the Fokker-Planck equation~\eqref{eq:FP-exp-re}.
More precisely, we have
\begin{equation*}
\Big(\int_0^\infty e^{-2\tau}\|G_0^{N,m_0}(N\tau)-G_0(\tau)\|_{L^2(M\,dv)}^2\,d\tau\Big)^\frac12 \,\lesssim\,N^{-\frac1{12}}\|\<(\nabla_{v_0},v_0)\>^{4m_0+10}\mathfrak G\|_{L^2(M\, dv)},
\end{equation*}
and this bound can be improved to $N^{-1}\|\<(\nabla_{v_0},v_0)\>^{4m_0+21}\mathfrak G\|_{L^2(M\,dv)}$ if $d\ge 28m_0+147$.

\begin{eqnarray}
A_0(v)&:=&(B_0\ast M)(v),\label{eq:coeff-LB-re}\\
B_0(v)&:=&\frac{\Lambda_\Vc}{|v|}\Big(\Id-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_\Vc\nabla^2|v|,\nonumber
\end{eqnarray}

\begin{equation}\label{eq:FN-diag-rem}
F_N\,:=\,
N\big(\tfrac{i}{\sqrt N}\big)^{2\ell+m+2}~{\tiny\begin{tikzpicture}[baseline={([yshift=-3ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (0) at (-0.5,0) {};
\node (1) at (0.5,0) {};
\node (2) at (0,0.6) {};
\node (3) at (2,0) {};
\node (4) at (1.5,0.6) {};
\node (5) at (2.5,0) {};
\node (6) at (-1,-0.5) {};
\node (7) at (-3,-1.3) {};
\end{scope}
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0) -- (1);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (1) -- (3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (3) -- (5);
\draw (0) -- (-0.5,0.6);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (-0.5,0.6) -- (2);
\draw (2) -- (0,0.3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0,0.3) -- (0.5,0.3);
\draw (0.5,0.3) -- (1);
\draw (4) -- (1.5,0.3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (1.5,0.3) -- (2,0.3);
\draw (2,0.3) -- (3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (2) -- (4);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (4) -- (2.5,0.6);
\draw (2.5,0.6) -- (5);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0) -- (-1,0);
\draw (-1,0) -- (6);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (2.5,-0.5) -- (6);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (6) -- (-1.5,-0.5);
\draw (-1.5,-0.5) -- (-1.5,-0.65);
\draw[dotted] (-1.5,-0.5) -- (-1.5,-1);
\draw (7) -- (-3,-1.1);
\draw[dotted] (-3,-1.1) -- (-3,-0.75);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (7) -- (2.5,-1.3);
\node at (1,0.3) {\ldots};
\node at (1,0.8) {($\ell$ times)};
\node at (-2.3,-0.5) {($m$ times)};
\end{tikzpicture}}~\Lc g^{N,m_0}_m.
\end{equation}

\begin{equation}\label{eq:coeff-LB-re-LB!}
B(v,w)=\int_{\R^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\e(k,k\cdot v)|^2}\,\frac{dk}{(2\pi)^{d}}.
\end{equation}

\begin{equation}\label{eq:orthogonality0}
\int_{\T^d\times\R^d}G_{N,m}(z_0,\ldots,z_m)\,M(z_j)\,dz_j=0,\qquad\text{for all $1\le j\le m$},
\end{equation}

\begin{eqnarray}
A(v)&:=&\int_{\R^d}B(v,v-v_*)M(v_*)\,dv_*,\label{eq:coeff-LB}\\
B(v,w)&:=&\sum_{k\in2\pi\Z^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\e(k,k\cdot v)|^2},\nonumber\\
\e(k,k\cdot v)&:=&1+\V(k)\int_{\R^d}\frac{k\cdot \nabla M(v_*)}{k\cdot(v-v_*)-i0}\,dv_*.\nonumber
\end{eqnarray}

\subsection{General overview}
Consider the dynamics of a tagged particle (labeled `$0$') in a system of $N+1$ interacting particles in the $d$-dimensional box $\mathbb T^d=[-\frac12,\frac12]^d$ with periodic boundary conditions, as described by Newton's equations
\begin{equation*}
\left\{\begin{array}{l}
\tfrac{d}{dt}X_{j}=V_j,\\[2mm]
\tfrac{d}{dt}V_{j}=-\frac1N\sum_{0\le l\le N}\nabla\Vc(X_j-X_l),\qquad0\le j\le N,\\[2mm]
(X_{j},V_j)|_{t=0}=(X_j^\circ,V_j^\circ),
\end{array}\right.
\end{equation*}
where $\{Z_j:=(X_j,V_j)\}_{0\le j\le N}$ is the set of positions and velocities of the particles in the phase space $\mathbb T^d\times\mathbb R^d$, where $\Vc:\mathbb T^d\to\mathbb R$ is a (long-range) interaction potential, and where the mean-field scaling is considered. For simplicity, we focus throughout this work on the case of a smooth potential~$\Vc\in C^\infty_c$.
In terms of a probability density $F_N$ on the $N$-particle phase space $(\mathbb T^d\times\mathbb R^d)^N$, this system of ODEs for trajectories leads formally to the Liouville equation
\begin{equation}\label{eq:Liouville}
\left\{\begin{array}{l}
\partial_tF_N+\sum_{0\le j\le N}v_j\cdot\nabla_{x_j}F_N\,=\,\frac1N\sum_{0\le j, l\le N}\nabla\Vc(x_j-x_l)\cdot\nabla_{v_j}F_N,\\[1mm]
F_N|_{t=0}\,=\,F_N^\circ.
\end{array}\right.
\end{equation}
For this system, we aim to study one of the predictions of the Lenard-Balescu theory, that is, the slow thermalization of the tagged particle `$0$' when the background particles are initially at thermal equilibrium. For simplicity, we assume that the tagged particle has initially spatially-homogeneous distribution: this ensures that the mean-field force on the tagged particle vanishes, so that thermalization becomes the leading effect. In other words, we choose initially
\[F_N^\circ(z_0,\ldots,z_N)\,=\,f^\circ(v_0)M_{N}(z_1,\ldots,z_N),\qquad z_j=(x_j,v_j),\]
where $f^\circ\in \mathcal{P}(\mathbb R^d)$ is the initial velocity density of the tagged particle
and where $M_{N}$ stands for the Gibbs equilibrium
\[M_{N}(z_1,\ldots,z_N)\,:=\,Z_{N}^{-1}\exp\bigg(-\frac\beta2\sum_{1\le j\le N}|v_j|^2-\frac\beta{2N}\sum_{1\le j,l\le N\atop j\ne l}\Vc(x_j-x_l)\bigg),\]
for some fixed inverse temperature $\beta\in(0,\infty)$ and normalization factor $Z_{N}$.
We emphasize that background particles are exchangeable
and that the system is invariant under spatial translations.
In this setting, due to interactions with the background, the Lenard-Balescu theory predicts that the tagged particle should slowly thermalize on timescales $t\gg N$ and progressively acquire a Maxwellian velocity distribution as the background itself,
\[M(v)\,:=\,(\tfrac\beta{2\pi})^\frac d2e^{-\frac\beta2|v|^2}.\]
More precisely, focussing on the velocity distribution of the tagged particle
\[f_{N,0}(t,v)\,:=\,\int_{\mathbb T^d\times(\mathbb T^d\times\mathbb R^d)^N}F_{N}(t,x,v,z_1,\ldots,z_N)\,dxdz_1\ldots dz_N,\]
it is predicted in~\cite{Piasecki-81,PS-87,DSR-21} (see also~\cite{RR-60,TGM-64} for the corresponding test particle problem without back-reaction) that the time-rescaled density $f_{N,0}(N\tau,v)$ should converge as $N\uparrow\infty$ to the solution~$f(\tau,v)$ of the linearized Lenard-Balescu equation at Maxwellian equilibrium (without loss term), which takes form of the following Fokker-Planck equation,
\begin{equation}\label{eq:FP-exp}
\left\{\begin{array}{l}
\partial_\tau f=\Div_v(A (\nabla+\beta v)f),\\
f|_{\tau=0}=f^\circ,
\end{array}\right.
\end{equation}
with diffusion tensor field $A$ given by the (periodic) Lenard-Balescu formula
\begin{eqnarray}
A(v)&:=&\int_{\mathbb R^d}B(v,v-v_*)M(v_*)\,dv_*,\label{eq:coeff-LB}\\
B(v,w)&:=&\sum_{k\in2\pi\mathbb Z^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\varepsilon(k,k\cdot v)|^2},\nonumber\\
\varepsilon(k,k\cdot v)&:=&1+\V(k)\int_{\mathbb R^d}\frac{k\cdot \nabla M(v_*)}{k\cdot(v-v_*)-i0}\,dv_*.\nonumber
\end{eqnarray}
In this formula, the dispersion function $\varepsilon$ modulating the collision kernel $B$ accounts for collective screening effects at equilibrium.
This Fokker-Planck equation~\eqref{eq:FP-exp} quantifies the thermalization
\[f(\tau,v)\to M(v)\qquad\text{as $\tau=N^{-1}t\uparrow\infty$.}\]
A rigorous justification is still beyond reach at the moment, and we refer to~\cite{VW-18,Winter-21,DSR-21,MD-21} for some partial results on the topic. To date, the best result is the consistency obtained in~\cite{DSR-21}, which only holds at best for relatively short times $\tau=N^{-1}t\ll N^{-3/4}$, thus missing the thermalization timescale $\tau\sim1$. We emphasize that the difficulty includes understanding the emergence of irreversibility, which here would only occur on long times as a fluctuation around the (trivial) mean-field behavior. In the present work, we introduce a simplified, hierarchical toy model for which a full derivation can be achieved, which will shed some new light on the problem.

\subsection{BBGKY approach}
To motivate our simplified model, we start by briefly recalling the standard BBGKY framework for addressing the above thermalization problem.
Intuitively, the slow relaxation of the tagged particle arises from the nontrivial correlations it develops with the background. To capture these effects, we introduce a suitable notion of correlation functions.
For $0\le m\le N$, we first define the joint density of the tagged particle with $m$ background particles as the following marginal of the $N$-particle density $F_N$,
\[F_{N,m}(z_0,\ldots,z_m)\,:=\,\int_{(\mathbb T^d\times\mathbb R^d)^{N-m}}F_N(z_0,\ldots ,z_N)\,dz_{m+1}\ldots dz_N,\]
which is symmetric in its last $m$ variables $\{z_j\}_{1\le j\le m}$. Next, for $0\le m\le N$, we define the $m$-th order correlation function (or cumulant) by
\begin{equation}\label{eq:def-CNm}
G_{N,m}(z_0,\ldots,z_m)\,:=\,\sum_{j=0}^m(-1)^{m-j}\sum_{\sigma\in \mathcal S_j^m}\frac{F_{N,j}}{M^{\otimes j+1}}(z_0,z_\sigma),
\end{equation}
where $\mathcal S_j^m$ stands for the set of all subsets of $\{1,\ldots,m\}$ with $j$ elements and where we use the short-hand notation $z_\sigma=(z_{i_1},\ldots,z_{i_j})$ for $\sigma=\{i_1,\ldots,i_j\}$. This definition ensures the orthogonality property
\begin{equation}\label{eq:orthogonality0}
\int_{\mathbb T^d\times\mathbb R^d}G_{N,m}(z_0,\ldots,z_m)\,M(z_j)\,dz_j=0,\qquad\text{for all $1\le j\le m$},
\end{equation}
and also ensures that marginals can be recovered by means of a (linear) cluster expansion,
\begin{equation}\label{eq:cluster}
F_{N,m}(z_0,\ldots,z_m)\,=\,M^{\otimes m+1}(z_0,\ldots,z_m)\sum_{j=0}^m\sum_{\sigma\in\mathcal S_j^m}G_{N,j}(z_0,z_\sigma),\qquad\text{for all $0\le m\le N$}.
\end{equation}
By orthogonality~\eqref{eq:orthogonality0}, we deduce that correlation functions satisfy
\begin{equation}\label{eq:unitarity0}
\sum_{m=0}^N\binom{N}{m}\int_{(\mathbb T^d\times\mathbb R^d)^{m+1}}|G_{N,m}|^2M^{\otimes m+1}\,=\,\int_{(\mathbb T^d\times\mathbb R^d)^{N+1}}\frac{|F_N|^2}{M^{\otimes N+1}}.
\end{equation}
The right-hand side in this identity would be a conserved quantity if $M^{\otimes N+1}$ were replaced by the Gibbs ensemble~$M_{N+1}$. Still, it can be checked to be uniformly controlled in time: as shown in~\cite[Lemma~2.2]{DSR-21}, in the spirit of~\cite{BGSR-17}, we can deduce for all $0\le m\le N$ and $t\ge0$,
\begin{equation}\label{eq:apriori-CNm}
\Big(\int_{(\mathbb T^d\times\mathbb R^d)^{m+1}}|G_{N,m}(t)|^2M^{\otimes m+1}\Big)^\frac12\,\lesssim\,\binom{N}{m}^{-\frac12}\,\lesssim\,m^{\frac m2}N^{-\frac m2}.
\end{equation}
This can be viewed as some form of chaos estimates, showing that higher-order correlations are tinier in the limit $N\uparrow\infty$. Yet, the scaling in $N$ is not expected to be optimal: in particular, $G_{N,1}$ is expected to be only $O(N^{-1})$ instead of $O(N^{-1/2})$, cf.~\cite{DSR-21}.

To get more precise estimates on correlations, we need to go back to the Liouville equation~\eqref{eq:Liouville} for~$F_N$. Inserting the cluster expansion~\eqref{eq:cluster}, we check that correlations satisfy a hierarchy of equations of the following form, for $0\le m\le N$,
\begin{equation}\label{eq:hierarchy-true}
\partial_tG_{N,m}+iL_{N,m}G_{N,m}=iS^+_{N,m}G_{N,m+1}+\tfrac1N\Big(iS^\circ_{N,m}G_{N,m}+iS^-_{N,m}G_{N,m-1}+iS^=_{N,m}G_{N,m-2}\Big),
\end{equation}
where for notational convenience we set $G_{N,m}\equiv0$ for $m>N$ or $m<0$, where the operators~$S^\square_{N,m}$
are viewed as some `creation/annihilation' operators on the space of correlations,
\begin{equation*}
\begin{array}{rllll}
S_{N,m}^+&:&L^2(M)^{\otimes m+1}&\to&L^2(M)^{\otimes m}\\
S_{N,m}^\circ&:&L^2(M)^{\otimes m}&\to&L^2(M)^{\otimes m}\\
S_{N,m}^-&:&L^2(M)^{\otimes m-1}&\to&L^2(M)^{\otimes m},\\
S_{N,m}^=&:&L^2(M)^{\otimes m-2}&\to&L^2(M)^{\otimes m},
\end{array}
\end{equation*}
and where~$iL_{N,m}$ stands for the $m$-particle linearized Vlasov operator
\begin{multline}\label{eq:lin-mf0}
iL_{N,m}G_{N,m}\,=\,\sum_{j=0}^mv_j\cdot\nabla_{x_j}G_{N,m}\\[-3.5mm]
+\tfrac{N+1-m}N\sum_{j=1}^m\beta v_j\cdot\int_{\mathbb T^d\times\mathbb R^d}\nabla \Vc(x_j-x_{*})G_{N,m}(z_{[0,m]\setminus \{j\}},z_*)\,M(v_{*})\,dz_{*}.
\end{multline}
We refer to~\cite[Lemma~2.4]{DSR-21} for a detailed formulation of this hierarchy.

If the a priori correlation estimates~\eqref{eq:apriori-CNm} were known to hold in a stronger, smooth topology, a direct analysis of the above hierarchy would readily yield the expected kinetic limit $f_{N,0}(N\tau,v)\to f(\tau,v)$; see~\cite{DSR-21}.
However, due to filamentation effects in phase space, correlations become increasingly oscillatory over time and are only controlled a priori in $L^2$. This leads to possible resonances that may, in principle, prevent convergence to the kinetic limit. Consequently, in~\cite{DSR-21}, we only managed to establish a partial consistency result, valid on some intermediate timescale~$t\sim N^r$ with $r<\frac1{18}$ (which could be improved at best to~$r<\frac14$).

This motivates a more refined analysis of the hierarchy~\eqref{eq:hierarchy-true}, with the goal of tracking the oscillatory structure of correlations in greater detail and demonstrating that resonances cannot occur. An important observation --- crucial to the present work but not exploited in earlier studies --- is that the operator~$S_{N,m}^+$ involves a velocity average: hence, while filamentation is a priori viewed as a problem, phase mixing may in fact be leveraged beneficially in some terms.

\subsection{A simplified hierarchy}\label{sec:simpl}
As a first step toward analyzing the exact hierarchy~\eqref{eq:hierarchy-true}, we introduce a simplified setting that preserves the essential features of the original system, but for which we are able to rigorously justify the expected thermalization.
We start with the following two convenient simplifications, which do not affect the physical relevance of the model:
\begin{enumerate}[(H1)]
\item\label{H1} To avoid specific resonance issues on the torus, we consider the problem on the whole space~$\mathbb R^d$ instead of the periodic box $\mathbb T^d$. Physically, this amounts to considering a system of $NL^d$ background particles in a rescaled box $[-\frac L2,\frac L2]^d$ with periodic boundary conditions, and taking the large-box limit $L\uparrow\infty$ as $N\uparrow\infty$. This does not change much in the system physically, but has a very important simplifying effect as Fourier variables become continuous.
\smallskip\item\label{H2} We include a small $O(\frac1N)$ diffusion in velocity in the particle system. As this only acts on the slow relaxation timescale $t\sim N$, we do not expect it to change much to the problem, but it is useful to simplify the analysis.
\end{enumerate}
We introduce two additional simplifications, which we believe are not essential to our arguments but are highly convenient for streamlining the computations:
\begin{enumerate}[(H3)]
\item[(H3)]\label{H1} The linearized mean-field operator $iL_{N,m}$, cf.~\eqref{eq:lin-mf0}, is replaced by pure transport: this allows to avoid many technicalities and in particular to perform direct computations in Fourier space instead of appealing to linear Landau damping. Physically, this amounts to neglecting collective screening effects.
\smallskip\item[(H4)]\label{H2} In the exact hierarchy~\eqref{eq:hierarchy-true}, the creation and annihilation operators $S_{N,m}^\pm$ are not exact adjoints: they only become approximately so in the limit $N\uparrow\infty$. For finite $N$, the additional operators~$S_{N,m}^\circ$ and~$S_{N,m}^=$ correct this lack of adjointness, so as to ensure the approximate unitarity~\eqref{eq:unitarity0}. This reflects the fact that the Gibbs equilibrium $M_{N+1}$ differs from the tensorized mean-field ensemble~$M^{\otimes N+1}$ around which correlations are defined, cf.~\eqref{eq:def-CNm}. To simplify the structure, we replace $S_{N,m}^{\pm}$ by operators $S_m^\pm$ that are truly adjoint and are independent of $N$. The additional operators $S_{N,m}^\circ$ and~$S_{N,m}^=$ are then set to $0$. This leads to a hierarchy with a neater unitarity structure, in particular making~\eqref{eq:apriori-CNm} trivial.
\end{enumerate}
Finally, we introduce a last simplification, which plays a crucial role in our analysis and constitutes the main restriction of the present work:
\begin{enumerate}[(H5)]
\smallskip\item[(H5)]\label{H5} We truncate the hierarchy at a fixed order $m_0\ge1$ independently of $N$, thus setting $G_{N,m}\equiv0$ in~\eqref{eq:hierarchy-true} for all~$m>m_0$. This means that correlations of the tagged particle are restricted to involve at most $m_0$ background particles at once.
\end{enumerate}
As we shall see, truncating the hierarchy enables us to control the complexity of possible Feynman diagrams in the perturbative expansion. Nonetheless, the model remains highly nontrivial as it still leads to an infinite Dyson series; see Section~\ref{sec:diag}.

Let us introduce more precisely the simplified hierarchy that we will study. In view of~(H1), with the large-box limit, the phase space for the particles is now
\[\mathbb D:=\mathbb R^d\times\mathbb R^d\,\ni\,z=(x,v).\]
Recalling the symmetry, invariance, and orthogonality properties of correlation functions~\eqref{eq:def-CNm}, the state space for simplified correlations is similarly chosen as the direct sum
\[\mathcal H\,:=\,\bigoplus_{m=0}^\infty\Hc_m,\]
where $\Hc_m$ is the set of functions $G_m:\mathbb D^{m+1}\to\mathbb R:(z_0,\ldots,z_m)\mapsto G_m(z_0,\ldots,z_m)$ that are symmetric in their last $m$ variables $z_1,\ldots,z_m$, that are invariant under spatial translations $(z_0,\ldots,z_m)\mapsto(x_0+x,v_0,\ldots,x_m+x,v_m)$, $x\in\mathbb R^d$, and such that
\begin{gather*}
\int_{\mathbb D^{m+1}} \delta(x_0)\,|G_m(z_{[m]})|^2\,M^{\otimes m+1}(v_{[m]})\,dz_{[m]}<\infty,\\
\int_\mathbb D G_m(z_{[m]})\,M(v_j)\,dz_j=0,\quad\text{for all $1\le j\le m$,}
\end{gather*}
where we set for abbreviation $[m]:=\{0,\ldots,m\}$.
Recalling the unitarity structure~\eqref{eq:unitarity0} with $\binom{N}{m}\sim \frac{N^m}{m!}$ as $N\uparrow\infty$ for fixed $m$, we endow the space~$\Hc_m$ with the Hilbert norm
\begin{equation}\label{eq:defHm}
\|G_m\|_{\Hc_m}^2\,:=\,\langle G_m,G_m\rangle_{\Hc_m}\,:=\,\frac1{m!}\int_{\mathbb D^{m+1}}\delta(x_0)\,|G_m(z_{[m]})|^2\,M^{\otimes m+1}(v_{[m]})\,dz_{[m]},
\end{equation}
thus leading to the following norm on the direct sum~$\mathcal H$,
\[\|(G_m)_m\|_\mathcal H^2\,:=\,\sum_{m=0}^\infty\|G_m\|_{\Hc_m}^2.\]
By spatial homogeneity, setting $G_0(v_0)\equiv G_0(z_0)$, we emphasize that $\Hc_0$ is identified with the weighted space~$L^2(M\,dv)$.
In this setting, for all $m\ge0$, we consider the skew-adjoint transport operator $iL_m$ and the self-adjoint velocity-diffusion operator $D_m$ on $\Hc_m$,
\begin{equation}\label{eq:mainop}
iL_m:=\sum_{0\le j\le m}v_j\cdot\nabla_{x_j},\qquad
D_m:=-\sum_{0\le j\le m}(\nabla_{v_j}-\tfrac\beta2 v_j)^2.
\end{equation}
Next, only keeping the leading contributions in the actual BBGKY creation and annihilation operators described e.g.\@ in~\cite{DSR-21}, we define
\begin{eqnarray}
iS_m^-G_{m-1}(z_{[m]})&:=&\sum_{0\le j\le m}\sum_{1\le l\le m\atop l\ne j}\nabla\Vc(x_j-x_l)\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m-1}(z_{[m]\setminus\{l\}}),\label{eq:def-Sm+-}\\
iS_{m-1}^+G_{m}(z_{[m-1]})&:=&\sum_{0\le j\le m-1}\int_\mathbb D\nabla\Vc(x_j-x_{m})\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\,dz_{m},\nonumber
\end{eqnarray}
in terms of the interaction potential $\Vc\in C^\infty_c(\mathbb R^d)$.
These simplified operators have the advantage of satisfying the exact adjointness relation
\begin{equation}\label{eq:adjS}
\langle H_{m-1},S_{m-1}^+G_{m}\rangle_{\Hc_{m-1}}=\langle S_{m}^-H_{m-1}, G_{m}\rangle_{\Hc_m},\qquad m\ge1.
\end{equation}
Instead of~\eqref{eq:hierarchy-true},
letting $m_0$ denote the maximal order of correlations, cf.~(H5), and including an $O(\frac1N)$ diffusion in velocity, cf.~(H2), we consider the following simplified hierarchy of equations on $\mathcal H$,
\begin{equation}\label{eq:hierarchy-simpl}
\fbox{\text{$\partial_tG_m^{N,m_0}+(iL_m+\tfrac\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\tfrac1N iS_m^-G_{m-1}^{N,m_0},\qquad 0\le m\le m_0,$}}
\end{equation}
where we set $G_{m}^{N,m_0}\equiv0$ for $m>m_0$ or $m<0$, and where we let $\kappa>0$ be some diffusion constant. Regarding initial data, we set
\begin{equation}\label{eq:hierarchy-simpl-ci}
G_m^{N,m_0}|_{t=0}\,=\,\left\{\begin{array}{lll}
\mathfrak G&:&m=0,\\
0&:&m\ne0,
\end{array}\right.
\end{equation}
for some initial velocity density $\mathfrak G\in\Hc_0$.
While in the original hierarchy~\eqref{eq:hierarchy-true} we have $m_0=N\uparrow\infty$, we consider here a fixed truncation parameter $m_0$ independently of $N\uparrow\infty$.

By the exact adjointness relation~\eqref{eq:adjS} for $S_m^\pm$,
recalling that the transport operator $iL_m$ is skew-adjoint and that $D_m$ is nonnegative, we directly find
\[\frac{d}{dt}\sum_{m=0}^{m_0}N^{m}\|G_m^{N,m_0}\|_{\Hc_m}^2\,=\,-\frac{2\kappa}N\sum_{m=0}^{m_0}N^{m}\sum_{j=1}^m\|(\nabla_{v_j} -\tfrac\beta2v_j)\,G_m^{N,m_0}\|_{\Hc_m}^2\,\le\,0.\]
Hence, for all $0\le m\le m_0$ and $t\ge0$, we deduce
\begin{equation}\label{eq:apriori-GNm-mod}
\|G_m^{N,m_0}(t)\|_{\Hc_m}\,\le\,N^{-\frac m2}\|\mathfrak G\|_{\Hc_0},
\end{equation}
which is the analogue of~\eqref{eq:apriori-CNm} in the present simplified setting.

Similarly as for the original hierarchy~\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\tau,v)$ is now expected to converge weakly as $N\uparrow\infty$ to the solution $G_0(\tau,v)$ of the following Fokker-Planck equation,
\begin{equation}\label{eq:FP-exp-re}
\left\{\begin{array}{l}
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2v)\cdot (\kappa\operatorname{Id}+A_0)(\nabla_{v}-\tfrac\beta 2v)G_0,\\
G_0|_{\tau=0}=\mathfrak G,
\end{array}\right.
\end{equation}
where the diffusion tensor field $A_0\ge0$ is given by the Landau formula
\begin{eqnarray}
A_0(v)&:=&(B_0\ast M)(v),\label{eq:coeff-LB-re}\\
B_0(v)&:=&\frac{\Lambda_\Vc}{|v|}\Big(\operatorname{Id}-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_\Vc\nabla^2|v|,\nonumber
\end{eqnarray}
with the explicit prefactor
$\Lambda_\Vc:=\frac{\omega_{d-1}}{d\omega_d}\int_{\mathbb R^d}|k|\pi\V(k)^2\frac{dk}{(2\pi)^d}$,
where $\omega_n$ stands for the volume of the $n$-dimensional unit ball.

\begin{rem}[From Lenard-Balescu to Landau kernel]
Formula~\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:
\begin{enumerate}[---]
\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\eqref{eq:coeff-LB} is replaced by its continuum version
\begin{equation}\label{eq:coeff-LB-re-LB!}
B(v,w)=\int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\varepsilon(k,k\cdot v)|^2}\,\frac{dk}{(2\pi)^{d}}.
\end{equation}
\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,
\begin{equation}\label{eq:comput-Landau}
B(v,w)\leadsto \int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\delta(k\cdot w)\,\frac{dk}{(2\pi)^{d}}\,=\,\frac{\Lambda_\Vc}{|w|}\Big(\operatorname{Id}-\frac{w\otimes w}{|w|^2}\Big).
\end{equation}
\item As we have introduced $O(\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\eqref{eq:FP-exp-re}.
\end{enumerate}
\end{rem}
\begin{rem}[Modified equilibrium structure]\label{rem:mod-equ}
Beyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\eqref{eq:FP-exp-re} also differs slightly from that in~\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\eqref{eq:FP-exp-re} describes relaxation to~$\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.
\end{rem}

\subsection{Main result}
In the framework of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\eqref{eq:FP-exp-re} for the tagged particle density.

Similarly as for the original hierarchy~\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\tau,v)$ is now expected to converge weakly as $N\uparrow\infty$ to the solution $G_0(\tau,v)$ of the following Fokker-Planck equation,
\begin{equation}\label{eq:FP-exp-re}
\left\{\begin{array}{l}
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2v)\cdot (\kappa\operatorname{Id}+A_0)(\nabla_{v}-\tfrac\beta 2v)G_0,\\
G_0|_{\tau=0}=\mathfrak G,
\end{array}\right.
\end{equation}
where the diffusion tensor field $A_0\ge0$ is given by the Landau formula
\begin{eqnarray}
A_0(v)&:=&(B_0\ast M)(v),\label{eq:coeff-LB-re}\\
B_0(v)&:=&\frac{\Lambda_\Vc}{|v|}\Big(\operatorname{Id}-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_\Vc\nabla^2|v|,\nonumber
\end{eqnarray}
with the explicit prefactor
$\Lambda_\Vc:=\frac{\omega_{d-1}}{d\omega_d}\int_{\mathbb R^d}|k|\pi\V(k)^2\frac{dk}{(2\pi)^d}$,
where $\omega_n$ stands for the volume of the $n$-dimensional unit ball.

\begin{rem}[From Lenard-Balescu to Landau kernel]
Formula~\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:
\begin{enumerate}[---]
\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\eqref{eq:coeff-LB} is replaced by its continuum version
\begin{equation}\label{eq:coeff-LB-re-LB!}
B(v,w)=\int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\varepsilon(k,k\cdot v)|^2}\,\frac{dk}{(2\pi)^{d}}.
\end{equation}
\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,
\begin{equation}\label{eq:comput-Landau}
B(v,w)\leadsto \int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\delta(k\cdot w)\,\frac{dk}{(2\pi)^{d}}\,=\,\frac{\Lambda_\Vc}{|w|}\Big(\operatorname{Id}-\frac{w\otimes w}{|w|^2}\Big).
\end{equation}
\item As we have introduced $O(\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\eqref{eq:FP-exp-re}.
\end{enumerate}
\end{rem}
\begin{rem}[Modified equilibrium structure]\label{rem:mod-equ}
Beyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\eqref{eq:FP-exp-re} also differs slightly from that in~\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\eqref{eq:FP-exp-re} describes relaxation to~$\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.
\end{rem}

\subsection{Main result}
In the framework of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\eqref{eq:FP-exp-re} for the tagged particle density.

\subsection{Main result}
In the framework of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\eqref{eq:FP-exp-re} for the tagged particle density.

\subsection{Dyson series expansion}\label{sec:diag}
Applying Laplace transform in the form~\eqref{eq:Lap}, the simplified hierarchy~\eqref{eq:hierarchy} reads as follows,
\begin{equation}\label{eq:Duhamel}
\left\{\begin{array}{l}
\big(1+i\alpha+\kappa\tfrac{t_N}{N}\hat D_0\big)\Lc g_0^{N,m_0}=\mathfrak g+\tfrac{t_N}{\sqrt N}i\hat S_0^+\Lc g_{1}^{N,m_0},\\[2mm]
\big(\tfrac{1+i\alpha}{t_N}+i\hat L_m+\tfrac\kappa N\hat D_m\big)\Lc g_m^{N,m_0}=\tfrac1{\sqrt N}\Big(i\hat S_m^+\Lc g_{m+1}^{N,m_0}+i\hat S_m^-\Lc g_{m-1}^{N,m_0}\Big),\qquad 1\le m\le m_0.
\end{array}\right.
\end{equation}
Iteratively solving this hierarchy, we obtain an infinite Dyson series for each correlation function $g_m^{N,m_0}$. Each term in this expansion consists of a sequence of resolvents
\[(\tfrac{1+i\alpha}{t_N}+i\hat L_m+\tfrac\kappa N\hat D_m)^{-1},\]
interlaced with creation or annihilation operators $\hat S_m^\pm$, acting on the initial data $\mathfrak g$.
To organize these contributions, we introduce a representation using Feynman-type diagrams:
\begin{enumerate}[---]
\item Each connected horizontal line corresponds to a different particle, with the lower line corresponding to the tagged particle `$0$'.
\smallskip\item Parallel horizontal segments stand for free propagators,
\[\big(\tfrac{1+i\alpha}{t_N}+i\hat L_1+\tfrac\kappa N\hat D_1\big)^{-1}=
{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\draw (0,0) -- (1,0);
\draw (0,0.3) -- (1,0.3);
\end{tikzpicture}}\qquad
\big(\tfrac{1+i\alpha}{t_N}+i\hat L_2+\tfrac\kappa N\hat D_2\big)^{-1}=
{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\draw (0,0) -- (1,0);
\draw (0,0.3) -- (1,0.3);
\draw (0,0.6) -- (1,0.6);
\end{tikzpicture}}\qquad\text{etc.}\]
\item Vertical segments merging two horizontal levels are viewed as ``collisions'' and correspond to applying creation or annihilation operators,
\[\hat S^{m,+}_{j,l}
={\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\end{scope}
\draw (1) -- (0,0.5);
\draw[dotted] (-0.4,0) -- (1) -- (0.4,0);
\draw[dotted] (0,0.5) -- (0.4,0.5);
\node at (1.5,0.5) {\tiny (particle $l$)};
\node at (1.5,0) {\tiny (particle $j$)};
\end{tikzpicture}}\qquad
\hat S^{m,-}_{j,l}
={\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\end{scope}
\draw (1) -- (0,0.5);
\draw[dotted] (-0.4,0) -- (1) -- (0.4,0);
\draw[dotted] (0,0.5) -- (-0.4,0.5);
\node at (1.5,0.5) {\tiny (particle $l$)};
\node at (1.5,0) {\tiny (particle $j$)};
\end{tikzpicture}}\]
We emphasize that for~$\hat S_{j,l}^{m,-}$ the annihilated index $l$ only runs over $1\le l\le m$, meaning that the tagged particle `$0$' is never annihilated.
\smallskip\item Diagrams are arbitrary compositions of the above two ingredients, alternating between free propagators and collisions.
The \emph{complexity of a diagram} is the maximal number of horizontal levels that appear at once on top of the lower level: by assumption, it is always bounded by the truncation parameter~$m_0$. Viewing time as flowing from right to left, we call \emph{input} (resp.\@ \emph{output}) variables the list of particle indices that are present at the right (resp.\@ left) of the diagram.
\end{enumerate}
To give a concrete example,
\[{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\node (2) at (0.5,0) {};
\node (3) at (1,0.5) {};
\node (4) at (1.5,0) {};
\end{scope}
\draw (1) -- (2) -- (4) -- (1.5,0.5) -- (3) -- (0,0.5) -- (1);
\draw (2) -- (0.5,0.25) -- (1,0.25) -- (3);
\end{tikzpicture}}
\,=\, S_{0,1}^{0,+}\big(\tfrac{1+i\alpha}{t_N}+i\hat L_1+\tfrac\kappa N\hat D_1\big)^{-1}S_{0,2}^{1,+}\big(\tfrac{1+i\alpha}{t_N}+i\hat L_2+\tfrac\kappa N\hat D_2\big)^{-1}S_{1,2}^{2,-}\big(\tfrac{1+i\alpha}{t_N}+i\hat L_1+\tfrac\kappa N\hat D_1\big)^{-1}S_{0,1}^{1,-},\]
and thus, inserting the definition of the operators and carefully tracking the Fourier variables,
\begin{multline*}
{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\node (2) at (0.5,0) {};
\node (3) at (1,0.5) {};
\node (4) at (1.5,0) {};
\end{scope}
\draw (1) -- (2) -- (4) -- (1.5,0.5) -- (3) -- (0,0.5) -- (1);
\draw (2) -- (0.5,0.25) -- (1,0.25) -- (3);
\end{tikzpicture}}\,\mathfrak g(\alpha,v_0)
\,=\, \int_{\Dd^2}
\sqrt M(v_1)\,k_1\V(k_1)\cdot\nabla_{v_0}\Big(\tfrac{1+i\alpha}{t_N}+ik_1\cdot(v_1-v_0)-\tfrac\kappa N\triangle_{v_{[1]}}\Big)^{-1}\\
\times \sqrt M(v_2)\,k_2\V(k_2)\cdot\nabla_{v_0}\Big(\tfrac{1+i\alpha}{t_N}+ik_1\cdot(v_1-v_0)+ik_2\cdot(v_2-v_0)-\tfrac\kappa N\triangle_{v_{[2]}}\Big)^{-1}\\
\times \sqrt M(v_2)\,k_2\V(k_2)\cdot\nabla_{v_0}\Big(\tfrac{1+i\alpha}{t_N}+i(k_1+k_2)\cdot(v_1-v_0)-\tfrac\kappa N\triangle_{v_{[1]}}\Big)^{-1}\\
\times\sqrt M(v_1)\,(k_1+k_2)\V(k_1+k_2)\cdot\nabla_{v_0}\mathfrak g(v_0)\,d^*\hat z_1d^*\hat z_2,
\end{multline*}
where for an index set $A$ we let $\triangle_{v_A}=\sum_{j\in A}\triangle_{v_{j}}$.
Occasionally, we shall indicate the associated Fourier momentum variables above each horizontal segment, thus representing the above integral as
\[=\,\int_{(\R^d)^2}
{\tiny\begin{tikzpicture}[baseline={([yshift=-2ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\node (2) at (1.5,0) {};
\node (3) at (3,0.8) {};
\node (4) at (4.5,0) {};
\end{scope}
\draw (1) -- (2) -- (4) -- (4.5,0.8) -- (3) -- (0,0.8) -- (1);
\draw (2) -- (1.5,0.4) -- (3,0.4) -- (3);
\node at (0.75,0.17) {$-k_1$};
\node at (1.5,0.97) {$k_1$};
\node at (3,0.17) {$-k_1-k_2$};
\node at (2.25,0.57) {$k_2$};
\node at (3.75,0.97) {$k_1+k_2$};
\end{tikzpicture}}
\,\mathfrak g(v_0)\,d^*k_1d^*k_2.\]
Note that the definition of the creation and annihilation operators ensures the conservation of the sum of Fourier variables at each collision.

As detailed in Section~\ref{sec:strategy}, the proof relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:
\begin{enumerate}[---]
\item {\it Renormalization:} The Dyson series must be renormalized to eliminate the leading recollisions, which amounts to factoring out part of the expansion. Concretely, the free transport propagators around which the expansion is performed are replaced by renormalized propagators, obtained by adding to the free transport an approximate (non-Markovian) version of the limiting Fokker-Planck operator (so-called ``hat operator'' below).
\smallskip\item {\it Phase mixing:} Since annihilation operators involve velocity averages, cf.~\eqref{eq:def-Sm+-}, phase mixing can be effectively exploited in several terms of the expansion. In our approach, this is achieved by systematically applying contour deformations in all the integrals over free velocity variables.
\smallskip\item {\it Hypoelliptic estimates:} As explained, the renormalization effectively augments the free transport with a (non-Markovian) Fokker-Planck-type operator carrying a prefactor of order $O(\frac1N)$. By treating this, heuristically, as a genuine Fokker-Planck operator, filamentation effects in phase space are mitigated through the induced $O(\frac1N)$ velocity diffusion. This allows to replace naive resolvent estimates with hypoelliptic ones, which substantially improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, we exploit the additional $O(\frac1N)$ velocity diffusion included for simplicity in the system to justify these improved bounds.
\end{enumerate}
As discussed heuristically in Section~\ref{sec:limitation} below, the restriction~\eqref{eq:def de d_0} on the space dimension relative to the truncation parameter arises from geometric constraints in the complex deformations used for phase-mixing arguments and does not appear to be avoidable at present. Several refinements of the method could improve the value of $d_0$ in~\eqref{eq:def de d_0}, but none seem enough to remove the linear dependence on the truncation parameter.

\begin{rem}[Scope and limitations of the model]
Several of the simplifying assumptions in our model could be relaxed without substantial modifications. In particular, the free transport operator could be replaced by the linearized Vlasov operator appearing in the original hierarchy, cf.~\eqref{eq:lin-mf0}, in which case the Landau kernel in the formula~\eqref{eq:coeff-LB-re} for the diffusion tensor would be replaced by the Lenard-Balescu kernel~\eqref{eq:coeff-LB-re-LB!}. Likewise, the modification of the equilibrium structure noted in Remark~\ref{rem:mod-equ} could be removed with only minor adjustments.
A more consequential simplification of the model concerns the choice of creation and annihilation operators as exact adjoints; we expect that the approximate unitarity of the original hierarchy~\eqref{eq:hierarchy-true} would still suffice for the analysis, but this remains to be verified. Another technical simplification is the inclusion of an $O(\frac1N)$ velocity diffusion in the system, which we cannot yet dispense with. Finally, the most restrictive assumption is, of course, the truncation of the hierarchy itself, which currently appears unavoidable.
\end{rem}

\begin{rem}[Related work]
While completing this work, we became aware of an ongoing parallel investigation by Bodineau and Le Bris~\cite{BLB}, which addresses another simplified model in the direction of a rigorous justification of Lenard-Balescu thermalization. Specifically, they derive a similar linearized Landau equation for a tagged particle interacting via mean-field forces within an ideal Rayleigh gas --- namely, a background of particles that do not interact with each other, only indirectly through the tagged particle.
Their analysis follows a trajectorial approach, but our hierarchical framework could in principle be applied to their model as well. In that case, the diagrammatic structure is significantly simpler: since background particles do not interact, the creation and annihilation operators always correspond to collisions involving the tagged particle, and the pathological diagrams~\eqref{eq:err-ex} and~\eqref{eq:FN-diag-rem} that constitute the main difficulty in our analysis would be absent. In this simplified setting, we expect that the truncation of the hierarchy could be avoided, leading to a hierarchical proof of their result. We note, however, that our approach would still require the inclusion of an $O(\frac1N)$ velocity diffusion in the system, which is not needed in~\cite{BLB}.
\end{rem}

As detailed in Section~\ref{sec:strategy}, the proof of Theorem~\ref{th:main} relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:
\begin{enumerate}[---]
\item \textit{Renormalization:} The Dyson series is renormalized to eliminate the leading recollisions, by \emph{factoring out part of the expansion}. Concretely, the free transport propagators are replaced by \emph{renormalized propagators}, obtained by adding to free transport an \emph{approximate (non-Markovian) version of the limiting Fokker-Planck operator} (the ``hat operator'').
\item \textit{Phase mixing:} Because annihilation operators involve velocity averages (cf.~\eqref{eq:def-Sm+-}), phase mixing is exploited in several terms of the expansion by \emph{systematically applying contour deformations} in the integrals over free velocity variables.
\item \textit{Hypoelliptic estimates:} Renormalization augments free transport with a (non-Markovian) Fokker-Planck-type operator with prefactor $O(\tfrac1N)$. Treating this heuristically as a genuine Fokker-Planck operator, filamentation effects are mitigated via the induced $O(\tfrac1N)$ velocity diffusion, allowing one to \emph{replace naive resolvent estimates with hypoelliptic ones} to improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, the argument \emph{exploits the additional $O(\tfrac1N)$ velocity diffusion included for simplicity in the system} to justify these improved bounds.
\end{enumerate}
It is also noted that the dimension restriction \eqref{eq:def de d_0} (relative to the truncation parameter) arises from \emph{geometric constraints in the complex deformations used for phase-mixing arguments} and "does not appear to be avoidable at present."

As detailed in Section~\ref{sec:strategy}, the proof of the main theorem relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:
\begin{enumerate}[---]
\item \textit{Renormalization:} The Dyson series is renormalized to eliminate the leading recollisions, by \emph{factoring out part of the expansion}. Concretely, the free transport propagators are replaced by \emph{renormalized propagators}, obtained by adding to free transport an \emph{approximate (non-Markovian) version of the limiting Fokker-Planck operator} (the ``hat operator'').
\item \textit{Phase mixing:} Because annihilation operators involve velocity averages (cf.
\begin{eqnarray}
iS_m^-G_{m-1}(z_{[m]})&:=&\sum_{0\le j\le m}\sum_{1\le l\le m\atop l\ne j}\nabla\Vc(x_j-x_l)\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m-1}(z_{[m]\setminus\{l\}}),\label{eq:def-Sm+-}\\
iS_{m-1}^+G_{m}(z_{[m-1]})&:=&\sum_{0\le j\le m-1}\int_\Dd\nabla\Vc(x_j-x_{m})\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\,dz_{m},\nonumber
\end{eqnarray}
), phase mixing is exploited in several terms of the expansion by \emph{systematically applying contour deformations} in the integrals over free velocity variables.
\item \textit{Hypoelliptic estimates:} Renormalization augments free transport with a (non-Markovian) Fokker-Planck-type operator with prefactor $O(\tfrac1N)$. Treating this heuristically as a genuine Fokker-Planck operator, filamentation effects are mitigated via the induced $O(\tfrac1N)$ velocity diffusion, allowing one to \emph{replace naive resolvent estimates with hypoelliptic ones} to improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, the argument \emph{exploits the additional $O(\tfrac1N)$ velocity diffusion included for simplicity in the system} to justify these improved bounds.
\end{enumerate}
It is also noted that the dimension restriction
\begin{equation}\label{eq:def de d_0}
d\ge d_0=\left\{\begin{array}{cl}
2,&\text{if $m_0=1$},\\
8,&\text{if $m_0=2$},\\
28m_0+70,&\text{if $m_0\ge3$},
\end{array}\right.\end{equation}
(relative to the truncation parameter) arises from \emph{geometric constraints in the complex deformations used for phase-mixing arguments} and "does not appear to be avoidable at present."

\label{th:main}
Fix a truncation parameter $m_0\ge1$ and
assume that:
\begin{enumerate}[---]
\item the space dimension is sufficiently large depending on $m_0$, in the sense that
\begin{equation}\label{eq:def de d_0}
d\ge d_0=\left\{\begin{array}{cl}
2,&\text{if $m_0=1$},\\
8,&\text{if $m_0=2$},\\
28m_0+70,&\text{if $m_0\ge3$},
\end{array}\right.\end{equation}
\item the diffusion constant $\kappa$ is sufficiently large, in the sense that $\kappa\ge C_{m_0}\int_{\mathbb R^d}|k|\V(k)\,dk$ for some constant~$C_{m_0}>0$ depending only on $m_0$. In case $m_0=1$, this condition can be dropped.
\end{enumerate}
Then, for any initial condition $\mathfrak G\in C_c^\infty(\mathbb R^d)$, the solution $(G_m^{N,m_0})_{0\le m\le m_0}$ of the simplified hierarchy
\begin{equation}\label{eq:hierarchy-simpl}
\fbox{\text{$\partial_tG_m^{N,m_0}+(iL_m+\tfrac\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\tfrac1N iS_m^-G_{m-1}^{N,m_0},\qquad 0\le m\le m_0,$}}
\end{equation}
with initial condition
\begin{equation}\label{eq:hierarchy-simpl-ci}
G_m^{N,m_0}|_{t=0}\,=\,\left\{\begin{array}{lll}
\mathfrak G&:&m=0,\\
0&:&m\ne0,
\end{array}\right.
\end{equation}
satisfies that the time-rescaled tagged-particle density
\[(\tau,v)\mapsto G_0^{N,m_0}(N\tau,v)\]
converges strongly in $L^2_\loc(\mathbb R^+;L^2(M\, dv))$ to the solution $G_0$ of the Fokker-Planck equation
\begin{equation}\label{eq:FP-exp-re}
\left\{\begin{array}{l}
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2v)\cdot (\kappa\Id+A_0)(\nabla_{v}-\tfrac\beta 2v)G_0,\\
G_0|_{\tau=0}=\mathfrak G,
\end{array}\right.
\end{equation}
More precisely, we have
\begin{equation*}
\Big(\int_0^\infty e^{-2\tau}\|G_0^{N,m_0}(N\tau)-G_0(\tau)\|_{L^2(M\,dv)}^2\,d\tau\Big)^\frac12 \,\lesssim\,N^{-\frac1{12}}\|\<(\nabla_{v_0},v_0)\>^{4m_0+10}\mathfrak G\|_{L^2(M\, dv)},
\end{equation*}
and this bound can be improved to $N^{-1}\|\<(\nabla_{v_0},v_0)\>^{4m_0+21}\mathfrak G\|_{L^2(M\,dv)}$ if $d\ge 28m_0+147$.

Question:
Let $m_0\ge 1$. Assume the space dimension $d$ satisfies
\[
d\ge d_0:=\begin{cases}
2,& m_0=1,\\
8,& m_0=2,\\
28m_0+70,& m_0\ge 3,
\end{cases}
\]
and assume the diffusion constant $\kappa$ is large enough in the sense that
\[
\kappa\ge C_{m_0}\int_{\mathbb R^d}|k|\,\mathcal V(k)\,dk
\]
for some constant $C_{m_0}>0$ depending only on $m_0$ (this assumption is not needed when $m_0=1$). For any initial datum $\mathfrak G\in C_c^\infty(\mathbb R^d)$, let $(G_m^{N,m_0})_{0\le m\le m_0}$ solve
\[
\partial_t G_m^{N,m_0}+\Big(iL_m+\frac\kappa N D_m\Big)G_m^{N,m_0}
= iS_m^+G_{m+1}^{N,m_0}+\frac{i}{N}S_m^-G_{m-1}^{N,m_0},\qquad 0\le m\le m_0,
\]
with initial data
\[
G_m^{N,m_0}\big|_{t=0}=\begin{cases}
\mathfrak G,& m=0,\\
0,& m\ne 0.
\end{cases}
\]
Define the diffusion tensor
\[
A_0(v)=(B_0*M)(v),\qquad
B_0(v)=\frac{\Lambda_{\mathcal V}}{|v|}\Big(\mathrm{Id}-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_{\mathcal V}\nabla^2|v|,
\]
where
\[
\Lambda_{\mathcal V}:=\frac{\omega_{d-1}}{d\omega_d}\int_{\mathbb R^d}|k|\,\pi\,\mathcal V(k)^2\frac{dk}{(2\pi)^d},
\]
and let $G_0(\tau,v)$ solve the Fokker--Planck equation
\[
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2 v)\cdot (\kappa\mathrm{Id}+A_0)(\nabla_v-\tfrac\beta 2 v)G_0,
\qquad G_0\big|_{\tau=0}=\mathfrak G.
\]
Here $L^2(M\,dv)$ denotes the weighted space with measure $M(v)\,dv$. Which quantitative estimate holds for the rescaled tagged-particle density $(\tau,v)\mapsto G_0^{N,m_0}(N\tau,v)$?

The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, if in addition $d\ge 28m_0+147$, then this improves to
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]

Choices:
B.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, if in addition $d\ge 28m_0+70$, then this improves to
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
C.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
D.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, there exists a constant $C>0$ independent of $m_0$ such that, whenever
$d\ge 28m_0+147$ and $\kappa\ge C\int_{\mathbb R^d}|k|\,\mathcal V(k)\,dk$, the estimate improves to
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
E.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, if in addition $d\ge 28m_0+147$, then one has the same bound with the stronger norm $\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}$ on the right-hand side.

extra-dimension threshold for the $N^{-1}$ improvement

dropped the improved $N^{-1}$ conclusion under the extra dimension hypothesis

dependence of the largeness constant on $m_0$

baseline scaling improved from $N^{-1/12}$ to $N^{-1}$ without the extra high-dimensional hypothesis