\begin{thm}
\label{thm:main}
Let $\beta(n)\colonequals \max_{w\in S_n}|\supp(\mathfrak S_w)|$. Then
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)} = 1.
\]
More precisely,
\[
-\ln(4)-1 \leq \liminf_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq \limsup_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq -1.
\]
\end{thm}

Schubert polynomials $\mathfrak S_w(x_1, \dots, x_n)$, indexed by permutations $w\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \cite{ls82}. The specialization $\mathfrak S_w(1,\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\colonequals \max_{w\in S_n}\mathfrak S_w(1, \dots,1)$, Stanley \cite{stanley17} observed that
\[
\frac14 \leq \liminf_{n\to\infty} \frac{\log_2(u(n))}{n^2} \leq \limsup_{n\to\infty}\frac{\log_2(u(n))}{n^2} \leq \frac12
\]
and asked whether $\lim_{n\to\infty}\frac{\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.

We study the growth rate of the maximal sizes of \emph{supports} of Schubert polynomials. Write $\mathrm{supp}(\mathfrak S_w)\colonequals \{\alpha \in \ZZ_{\geq 0}^n\colon \mathbf x^\alpha\textup{ appears in } \mathfrak S_w\}$.

The key ingredient in our proof of Theorem~\ref{thm:main} is the observation that a certain \emph{layered} permutation has support of size asymptotically at least $n!/4^n$. Layered permutations have previously appeared in Merzon--Smirnov's conjecture \cite{ms16} that $\mathfrak S_w(1,\dots,1)$ is maximized at a layered permutation. Guo--Lin conjectured \cite[Prob 5.3]{gl24} that there exist permutations simultaneously maximizing $\mathfrak S_w(1, \dots,1)$ and $|\supp(\mathfrak S_w)|$.

We will use the following (crude) estimates for the factorial function, whose proofs we include for completeness.
\begin{lem}
\label{lem:factorial-estimate}
The inequalities
\[
e\left(\frac ne\right)^n \leq n! \leq e\left(\frac{n+1}e\right)^{n+1}
\]
hold for all positive integers $n$.
\end{lem}
\begin{proof}
Estimate $\ln(n!) = \sum_{k=1}^n\ln(k)$ by the integrals
\[
\int_1^n \ln(x)\,dx \leq \sum_{k=1}^n \ln(k) \leq \int_1^{n+1}\ln(x)\,dx.
\]
The definite integrals evaluate to $n\ln(n) - n + 1$ and $(n+1)\ln(n+1) - (n+1) + 1$. The claim follows.
\end{proof}
\begin{lem}
\label{lem:floor-fact-est}
The inequality
\[
\lfloor k\rfloor! \geq \frac1{2k}\left(\frac ke\right)^k
\]
holds for all real numbers $k>1$.
\end{lem}
\begin{proof}
Using Lemma~\ref{lem:factorial-estimate}, compute that
\[
\lfloor k\rfloor! \geq \frac1{k+1} \lceil k\rceil! \geq \frac1{k+1} \cdot \frac{\lceil k\rceil^{\lceil k\rceil}}{e^{\lceil k\rceil-1}} \geq \frac1{k+1}\cdot \frac{k^k}{e^k}.
\]
As $k > 1$, the claim follows.
\end{proof}
\section{A layered permutation with large Schubitope}
Our main technical result is as follows.
\begin{thm}
\label{thm:tech-main}
Let $w = w(b_1, \dots, b_m)$ be a layered permutation in $S_n$. If $b_m \leq n/2$, then
\[
|\supp(\mathfrak S_w)| \geq b_m!\cdot |\supp(\mathfrak S_{w'})|, \qquad \textup{ where } w'\colonequals w(b_1, \dots, b_{m-1})\in S_{n-b_m}.
\]
\end{thm}
Our proof of Theorem~\ref{thm:tech-main} will use the following lemma.
\begin{lem}
\label{lem:last-block}
Let $w\in S_n$ be a layered permutation of the form $w = w(1, \dots, 1, b_m)$ for $b_m \leq n/2$. For any subset $S\subseteq D(w)$, there exists a diagram $C$ satisfying $C\leq D(w)$ and $C\cap D(w) = S$.
\end{lem}
\begin{proof}
Each column $D_j$ of $D(w)$ is of the form $\{n-b_m + 1, \dots, n-c\}$ for some $c \geq 1$. As $|D_j| \leq b_m-1 < n-b_m$, it follows that for any subset $S_j$ of $D_j$ there exists a set $C_j\leq D_j$ with $C_j \cap D_j = S_j$. Applying this column by column to $D(w)$, the claim follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:tech-main}]
For each of the $b_m!$ many vectors $\alpha\in\ZZ_{\geq0}^{b_m}$ satisfying $(0,\dots,0) \leq \alpha \leq (b_m - 1, b_m - 2, \dots, 1, 0)$, we construct an embedding $i_\alpha\colon \mathcal S_{D(w')} \hookrightarrow \mathcal S_{D(w)}$ so that $\img(i_\alpha)\cap\img(i_{\alpha'}) = \emptyset$ for all $\alpha\neq\alpha'$.

For each $\alpha$ satisfying $(0,\dots,0)\leq\alpha\leq (b_m - 1, b_m - 2, \dots, 1)$, use Lemma~\ref{lem:last-block} to choose a diagram $C_\alpha \leq D(w_{\mathrm{LB}})$ whose weight satisfies $\wt(C_\alpha)_{n-b_i + j} = \alpha_j$. Then define the map
\begin{align*}
i_\alpha\colon \mathcal S_{D(w')}&\to \mathcal S_{D(w)}\\
\wt(C')&\mapsto\wt((C', C_\alpha));
\end{align*}
the map $i_\alpha$ is well-defined and injective because it is in fact the translation map $\gamma \mapsto \gamma +\wt(C_\alpha)$. Furthermore, any $\gamma' \in \mathcal S_{D(w')}$ satisfies $\gamma'_{n-b_m+j} = 0$ for all $j\geq 0$, so any vector $\gamma\in\img(i_\alpha)$ satisfies $\gamma_{n-b_m+j} = \alpha_j$. In particular, the $\img(i_\alpha)$ are disjoint subsets of $\mathcal S_{D(w)}$.
\end{proof}
\begin{cor}
\label{cor:upshot}
Fix an integer $n\geq 3$ and let $c\colonequals \max\{k \colon \lfloor n/2^k\rfloor \geq 1\}$ and $d \colonequals n - \sum_{k=1}^c \lfloor n/2^k\rfloor$. Let $w$ be the layered permutation with blocks of size
\[
\underbrace{1,\dots,1}_{d\textup{ many}}, \left\lfloor \frac n{2^c}\right\rfloor, \left\lfloor \frac n{2^{c-1}}\right\rfloor, \dots, \left\lfloor\frac n4\right\rfloor,\left\lfloor \frac n2\right\rfloor.
\]
Then
\[
|\supp(\mathfrak S_w)| \geq \prod_{k=1}^c\left\lfloor \frac n{2^k}\right\rfloor!.
\]
In particular,
\[
\ln(|\supp(\mathfrak S_w)|) \geq \left(n - \frac1{\ln(2)}\ln(n) - 2\right)\ln(n) - n\ln(4) - (n-2).
\]
\end{cor}
\begin{proof}
From the inequality
\[
\left\lfloor \frac n{2^k}\right\rfloor \leq \frac n{2^k} = \frac12\left(n - \sum_{i=1}^{k-1} \frac n{2^i}\right) \leq \frac12\left(n - \sum_{i=1}^{k-1} \left\lfloor\frac n{2^i}\right\rfloor\right),
\]
Theorem~\ref{thm:tech-main} may be repeatedly applied to the layered permutation $w$ to obtain
\[
|\supp(\mathfrak S_w)| \geq \prod_{k=1}^c\left\lfloor \frac n{2^k}\right\rfloor!.
\]
Lemma~\ref{lem:floor-fact-est} implies
\begin{align*}
\prod_{k=1}^c\left \lfloor \frac n{2^k}\right\rfloor! &\geq \prod_{k=1}^c\left(\frac{2^{k-1}}n \left(\frac n{2^k e}\right)^{\frac n{2^k}}\right) \\&\geq \frac 1{n^c} \cdot \underbrace{\prod_{k=1}^c \left(\frac1{2^k}\right)^{\frac n{2^k}}}_{\geq 2^{-2n}}\cdot \underbrace{\prod_{k=1}^c \left(\frac ne\right)^{\frac n{2^k}}}_{\geq \left(\frac ne\right)^{n-2}}\label{eqn:estimate}\tag{$*$}\\
&\geq \frac{n^{n-c-2}}{4^n\cdot e^{n-2}}\\&\geq \frac{n^{n - \log_2(n) - 2}}{4^n \cdot e^{n-2}}
\end{align*}
where the estimates in~\eqref{eqn:estimate} follow from
\[
\sum_{k=1}^c k\cdot \frac n{2^k} \leq \sum_{k=1}^\infty k \cdot \frac n{2^k} = 2n
\]
and
\[
\frac n2 + \dots + \frac n{2^c} = n - \frac n{2^c} \geq n-2.
\]
\end{proof}

\newtheorem*{thm:main}{Theorem~\ref{thm:main}}
\begin{thm:main}
Let $\beta(n)\colonequals \max_{w\in S_n}|\supp(\mathfrak S_w)|$. Then
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)} = 1.
\]
More precisely,
\[
-\ln(4)-1 \leq \liminf_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq \limsup_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq -1.
\]
\end{thm:main}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
By Lemma~\ref{lem:upper-bound}, the inequality
\[
\ln(|\supp(\mathfrak S_w)|) \leq \ln(n!) \leq (n+1)\ln(n+1) - n
\]
holds for all $w\in S_n$. By Corollary~\ref{cor:upshot}, there exists $w\in S_n$ with
\[
\ln(|\supp(\mathfrak S_w)|) \geq \left(n-\frac1{\ln(2)}\ln(n) - 2\right)\ln(n) - n\ln(4) - (n-2).
\]
The inequalities
\[
-\ln(4)-1 \leq \liminf_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq \limsup_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq -1
\]
follow.
\end{proof}

\begin{prop}
\label{prop:groth-precise}
Fix an integer $n$, let $k$ be the unique integer so that $\binom{k+1}2\leq n<\binom{k+2}2$, and let $b \colonequals n - \sum_{i=1}^k i$. Let $w$ be the layered permutation with blocks of size $1, 2, \dots, k, b$. Then
\[
|\supp(\mathfrak G_w)| \geq \frac{n!}{n^{k+1}}.
\]
In particular,
\[
\ln(|\supp(\mathfrak G_w)|) \geq n\ln(n) - n - (\sqrt{2n}+1)\ln(n)
\]
\end{prop}
\begin{proof}
Let $\mathbf c = \mathrm{wt}(D(w))$ denote the Lehmer code of $w$, and let $\mathbf d = \mathrm{wt}(\overline{D(w)})$. Proposition~\ref{prop:fireworks-grothendieck} implies that $[\mathbf c, \mathbf d] \subseteq \supp(\mathfrak G_w)$, so
\begin{equation}
\label{eqn:crude-grothendieck}
|\supp(\mathfrak G_w)| \geq \prod_{i=1}^n (d_i - c_i + 1).
\end{equation}

\newtheorem*{thm:groth}{Theorem~\ref{thm:groth}}
\begin{thm:groth}
Let $\beta^{\mathfrak G}(n)\colonequals\max_{w\in S_n}|\supp(\mathfrak G_w)|$. Then
\[
\lim_{n\to\infty}\frac{\ln(\beta^{\mathfrak G}(n)) - n\ln(n)}n = -1.
\]
\end{thm:groth}
\begin{proof}[Proof of Theorem~\ref{thm:groth}]
By Lemma~\ref{lem:upper-bound}, the inequality
\[
\ln(|\supp(\mathfrak G_w)|)\leq \ln(n!) \leq (n+1)\ln(n+1)-n
\]
holds for all $w\in S_n$. By Proposition~\ref{prop:groth-precise}, there exists $w\in S_n$ with
\[
\ln(|\supp(\mathfrak G_w)|) \geq n\ln(n) - n - (\sqrt{2n}+1)\ln(n).
\]
The claim follows.
\end{proof}

Theorem~\ref{thm:main} answers a problem \cite[Prob 5.5]{gl24} posed by Guo--Lin. We do not know if the limit $\lim_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n$ exists, nor do we have a conjecture for its value.

The key ingredient in our proof of Theorem~\ref{thm:main} is the observation that a certain \emph{layered} permutation has support of size asymptotically at least $n!/4^n$. Layered permutations have previously appeared in Merzon--Smirnov's conjecture \cite{ms16} that $\mathfrak S_w(1,\dots,1)$ is maximized at a layered permutation. Guo--Lin conjectured \cite[Prob 5.3]{gl24} that there exist permutations simultaneously maximizing $\mathfrak S_w(1, \dots,1)$ and $|\mathrm{supp}(\mathfrak S_w)|$.

Grothendieck polynomials $\mathfrak G_w$ are inhomogeneous deformations of Schubert polynomials; they are generating functions for (possibly nonreduced) pipe dreams of $w$. We are able to produce a layered permutation with support of size asymptotically at least $n!/e^{\sqrt{2n} \cdot \ln(n)}$, i.e., of size $n!$ up to a subexponential factor. In particular, the maximal sizes of supports of Grothendieck polynomials satisfy the following more precise asymptotics.
\begin{thm}
\label{thm:groth}
Let $\beta^{\mathfrak G}(n)\colonequals\max_{w\in S_n}|\mathrm{supp}(\mathfrak G_w)|$. Then
\[
\lim_{n\to\infty}\frac{\ln(\beta^{\mathfrak G}(n)) - n\ln(n)}n = -1.
\]
\end{thm}

The post-theorem introduction indicates that “the key ingredient in our proof of Theorem~\ref{thm:main} is the observation that a certain \emph{layered} permutation has support of size asymptotically at least $n!/4^n$.”

\label{thm:main}
Let $\beta(n)\colonequals \max_{w\in S_n}|\mathrm{supp}(\mathfrak S_w)|$. Then
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)} = 1.
\]
More precisely,
\[
-\ln(4)-1 \leq \liminf_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq \limsup_{n\to\infty}\frac{\ln(\beta(n)) - n\ln(n)}n \leq -1.
\],

Question:
For each positive integer $n$, let $S_n$ be the symmetric group on $n$ letters. For a permutation $w\in S_n$, let $\mathfrak S_w(x_1,\dots,x_n)$ denote its Schubert polynomial, and define its support by
\[
\operatorname{supp}(\mathfrak S_w)\coloneqq \{\alpha\in \mathbb Z_{\ge 0}^n : \mathbf x^\alpha \text{ appears in } \mathfrak S_w\}.
\]
Set
\[
\beta(n)\coloneqq \max_{w\in S_n} |\operatorname{supp}(\mathfrak S_w)|.
\]
Which asymptotic estimate holds for $\beta(n)$?

One has
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)}=1.
\]
More precisely,
\[
-\ln(4)-1\le \liminf_{n\to\infty}\frac{\ln(\beta(n))-n\ln(n)}{n}
\le \limsup_{n\to\infty}\frac{\ln(\beta(n))-n\ln(n)}{n}\le -1.
\]

Choices:
B.
One has
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)}=1.
\]
More precisely,
\[
-\ln(4)-1\le \liminf_{n\to\infty}\frac{\ln(\beta(n))-n\ln(n)}{n}
\le \limsup_{n\to\infty}\frac{\ln(\beta(n))-n\ln(n)}{n}\le 0.
\]
C.
One has
\[
\limsup_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)}=1.
\]
D.
One has
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n^2}=0.
\]
More precisely,
\[
\limsup_{n\to\infty}\frac{\ln(\beta(n))-n\ln(n)}{n}= -1.
\]
E.
One has
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))}{n\ln(n)}=1.
\]
More precisely,
\[
\lim_{n\to\infty}\frac{\ln(\beta(n))-n\ln(n)}{n}= -1.
\]

The stem gives only definitions of Schubert polynomials, support, and β(n), then asks for the asymptotic estimate. It does not reveal the correct asymptotic form or its refined constants.

The item is close to a theorem-recall question: it essentially asks for the correct asymptotic statement of β(n). However, it is not a pure tautology, since the options vary in strength, precision, and limiting form, so the student must distinguish among competing formulations.

Some reasoning is required to compare nearby asymptotic claims, especially the weaker true statement (C), the overstrengthened statement (E), and the slightly altered bound in (B). Still, the question mainly tests recognition of the precise theorem statement rather than generating a conclusion from intermediate facts.

The distractors are mathematically plausible and well targeted: one weakens the conclusion, one slightly relaxes a bound, one mismatches normalization, and one improperly upgrades liminf/limsup bounds to an exact limit. These reflect realistic asymptotic reasoning errors.

A solid asymptotic-discrimination MCQ with no answer leakage and strong distractors, but it leans toward precise theorem recognition rather than deeper generative reasoning.


\begin{theorem}
\label{thm:opt-iid-homo}
Assume {\rm(A1), (A2), (A3)}, and $d\ge 3$. Let $\bar a,\bar\psi$ be as in Theorem~\ref{thm:opt-quant-homo}. Suppose that $f\in C^{4,\alpha}(\bar\B_1)$ and $g\in C^{6,\alpha}(\partial\B_1)$ for some $\alpha\in (0,1)$. Then, there exists a $C^6(\bar\B_1)$-extension of $g$ such that the solutions  $u, \bar u$ of \eqref{eq:elliptic-dirich}, \eqref{eq:effective-ellip} satisfy
\[
\max_{x\in B_R}|u(x)-\bar u(\tfrac{x}{R})|
\lesssim
\left\{
  \begin{array}{rl}
  	R^{-3/2}\norm{\bar u}_{C^6(\bar\B_1)}\tx &\text{ when } d=3,\\
  	R^{-2}(\log R)\norm{\bar u}_{C^6(\bar\B_1)}\tx&\text{ when }d\ge 4.
  \end{array}
\right.
\]
An example of such an extension is $g=\bar u$.
\end{theorem}

\begin{remark}\label{rmk:optimality}
This proposition identifies $O(R^{-1})$ as the optimal homogenization rate unless $\bar z \equiv 0$, and so no improvement beyond $O(R^{-1})$ should be expected in general.
         We further expect that Proposition~\ref{lem:two-scale-exp} remains valid for environments with finite range of dependence, for the same reasons discussed in the comments following Theorem~\ref{thm:opt-quant-homo}.
\end{remark}

\begin{align}
&\lambda^k_j(x)=\lambda^k_j(x;\omega) := a_j(x)\left[v^k(x+e_j)-v^k(x-e_j)\right],\;\;
\bar{\lambda}^k_j(x) :=E_{\mb P}\left[\rho(x)\lambda^k_j(x)\right],\label{eq:def-lambda}\\
&\eta_j(x)=\eta_j(x;\omega)\; :=a_j(x)\left[\xi(x+e_j)-\xi(x-e_j)\right],
\quad\;\;\,
\bar{\eta}_j(x) :=E_{\mb P}\left[\rho(x)\eta_j(x)\right],\label{eq:def-eta}
\end{align}

\label{sec:intro}
In this paper, we consider the homogenization convergence rates of purely second-order non-divergence form difference operators in an independent and identically distributed (i.i.d.) random environment on $\Z^d$ for dimensions $d\ge 3$.

Large-scale behavior of random walks in i.i.d.\ environments has been intensively studied; see \cite{BoSz-02, OZ-04, MB-11, DreRam-14, Kumagai-14} and the references therein. Since the dynamics of random walks in random environments (RWRE) can be described by difference equations, stochastic homogenization of difference equations in random environments is crucial for understanding the large-scale behavior of RWRE.
For environments in which spectral-gap or Efron--Stein type inequalities are available, the strong concentration properties of the medium can be exploited to obtain quantitative homogenization results; see, for example, \cite{GNO-15, AL-17, GT-23}. In this paper, we show that, for non-divergence form difference operators in the i.i.d.\ setting, the reflection symmetry of the law of the environment can in fact be used to extract additional cancellations. Somewhat unexpectedly, this yields homogenization rates for the Dirichlet problem that are strictly better than the standard optimal rate $O(R^{-1})$ obtained in \cite{GT-23}. To achieve this, we require a more delicate analysis of the fluctuations of the homogenization error, relying in particular on a deeper understanding of higher-order correctors.

\subsection{Settings}
Let $\mb S_{d\times d}$ denote the set of real $d\times d$ positive definite diagonal matrices. A map
\[
\omega:\Z^d\to\mb S_{d\times d}
\] is called an {\it environment}, and the set of all such environments is denoted by $\Omega$.
Let $\mb P$ be a probability measure on $\Omega$ such that
\[
\left\{\omega(x)=\mathrm{diag}[\omega_1(x),\ldots, \omega_d(x)], \;x\in\Z^d\right\}
\]
are i.i.d. under $\mb P$. We denote the expectation with respect to $\mb P$ by $\mb E$ (or $E_{\mb P}$).

\begin{definition}\label{def:differences}
Let $\{e_1,\ldots,e_d\}$ be the canonical basis for $\R^d$, and let
\[
U:=\left\{e\in\Z^d:|e|=1\right\}=\left\{\pm e_1,\ldots,\pm e_d\right\}
\]
be the set of unit vectors in $\Z^d$. We define the difference operators $\nabla=(\nabla_e)_{e\in U}$  and
$\nabla^2= \mathrm{diag}[\nabla_1^2,\ldots,\nabla_d^2]$ by
\begin{equation}\label{eq:def-nabla}
\nabla_e u(x) := u(x+e)-u(x), \qquad
\nabla_i^2u(x) := u(x+e_i)+u(x-e_i)-2u(x)
\end{equation}
for $e\in U$ and $i\in\{1,\ldots,d\}$.
Clearly, $\nabla$ and $\nabla^2$ are linear operators.
\end{definition}

For $r>0$ and $y\in\R^d$, we write
\[
\B_r(y) := \left\{x\in\R^d: |x-y|<r\right\}, \qquad
B_r(y):=\B_r(y)\cap\Z^d
\]
to denote the continuous and discrete ball with center $y$ and radius $r$, respectively.
When $y=0$, we write $\B_r := \B_r(0)$ and $B_r := B_r(0)$ for simplicity.
For any $B\subset\Z^d$, its {\it discrete boundary} $\partial B$ and {\it discrete closure} $\bar B$ are defined as
\[
\partial B:=\left\{z\in\Z^d\setminus B: |z-x|=1 \text{ for some }x\in B\right\},\qquad \bar B := B\cup\partial B.
\]
By abuse of notation, we also use $\partial A$ and $\bar A$ to denote the usual continuous boundary and closure of $A\subset\R^d$, respectively, if there is no confusion.
The interior of $B_R$ is denoted by
\[
\inB{R} := \{x\in B_R\,:\,  |x-y| \ge 1 \text{ for all }y\in\partial\B_R\}.
\]
Note that for any $R\ge 1$, there holds
\[
\bar{\B}_{R-1}\cap\Z^d\subset\inB
R\subset B_R\subset\binB{R}\subset\bar\B_R\cap\Z^d\subset\bar B_R.
\]

 For $x\in\Z^d$, a {\it spatial shift} is defined by
 \[
 \theta_x:\Omega\to\Omega,\qquad \theta_x\omega := \omega(x+\cdot).
 \]
In a random environment $\omega\in\Omega$ in dimensions $d\ge 2$,  we consider the discrete {\it non-divergence form}   elliptic Dirichlet problem
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
u(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
where $f,g\in\R^{\bar\B_1}$ are sufficiently regular, and $\zeta\in \R^\Omega$ satisfies suitable integrability conditions.
As $R\to\infty$, it is expected that $\vert u - \bar u(\frac{\cdot}{R})\rvert$ converges to $0$, where $\bar u$ solves
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
Here, $D^2 \bar{u}$ denotes the Hessian matrix of $\bar{u}$, and $\bar a=\bar a(\mb P)\in\mb S_{d\times d}$ and $\bar\psi=\bar\psi(\mb P,\zeta)\in\R$ are {\it deterministic} and do not depend on the realization of the random environment; see Theorem~\ref{thm:QCLT} and Theorem~\ref{thm:opt-quant-homo} for formulas of $\bar{a}$ and $\bar{\psi}$.

\medskip

We remark that in the continuous PDE setting, one only needs a function $g\in C^{2,\alpha}(\partial\B_1)$ on the boundary $\partial\B_1$ to consider Dirichlet problems of types \eqref{eq:elliptic-dirich} and \eqref{eq:effective-ellip}. However, in the discrete setting,
the discrete and continuous boundaries do not agree, and therefore the function $g$ in $\eqref{eq:elliptic-dirich}$ should be extended to $\frac{1}{R}\partial\inB R\not\subset\partial\B_1$.  This discrepancy typically induces an error of order $\tfrac{1}{R}$ which automatically becomes the ceiling of the optimal homogenization error in the discrete setting,  cf.  \cite{GT-23}.
Indeed, this $O(R^{-1})$ rate is the optimal rate obtained in \cite{GT-23} for $d\geq 3$.
Whereas, when such a discretization issue is not present, e.g., in the continuous periodic setting \cite{GTY-19, ST-21, Spr-24},  one needs to go deeper into higher order correctors to justify the optimality of the rate $O(R^{-1})$ for the homogenization of the Dirichlet problem. We refer to \cite{CSS-20, FGP-24, SSZ-25} for recent developments in numerical homogenization of PDEs in non-divergence form.

\medskip

A goal of this article is to isolate and examine the homogenization rate that does {\bf not} originate from boundary discretization.
To this end, we assume that the boundary data $g$ is properly extended to be a function on $\frac{1}{R}\partial\inB R$ in the Dirichlet problem \eqref{eq:elliptic-dirich}.

\medskip

The non-divergence form difference equation \eqref{eq:elliptic-dirich} is used to describe random walks in a random environment (RWRE) in $\Z^d$. To be specific, for $x,y\in \Z^d$, we set
\begin{equation}\label{def:omegaxei}
\omega(x,x\pm e_i):=\frac{\omega_i(x)}{2\,\tr\omega(x)} \quad  \text{ for } i=1,\ldots, d,
\end{equation}
 and
 \[
 \omega(x,y) := 0 \quad \text{ if $|x-y|\neq 1$},
 \]
 that is, we normalize $\omega$ to get a transition probability.
 We note that the configuration of $\{\omega(x,y):x,y\in\Z^d\}$ is also called a {\it balanced environment} in the literature \cite{L-82,GZ-12,BD-14,DGR-15, BCDG-18,DG-19}.
\begin{definition}\label{def:rwre-discrete}
For each fixed $\omega\in\Omega$, the random walk  $(X_n)_{n\ge 0}$ in the environment $\omega$ with $X_0=x$ is a Markov chain  in $\Z^d$ with transition probability $P_\omega^x$ specified by
\begin{equation}\label{eq:def-RW}
P_\omega^x\left(X_{n+1}=z|X_n=y\right) := \omega(y,z).
\end{equation}
The {\it continuous-time} RWRE $(Y_t)_{t\ge 0}$ in $\omega$ is defined to be the Markov process on $\Z^d$ with jump rates $\omega(x,y)$ from $x$ to $y$. With abuse of notations, we still denote by $P_\omega^x$  the quenched law of $(Y_t)$ if there is no ambiguity from the context.
\end{definition}
The expectation with respect to $P_\omega^x$ is denoted by $E_\omega^x$. When the starting point of the random walk is $0$, we sometimes omit the superscript and write $P_\omega^0$ and $E_\omega^0$ as $P_\omega$ and $E_\omega$, respectively. Note that for random walks $(X_n)$ in an environment $\omega$, we have that
\begin{equation}\label{def:omegabar}
\bar\omega^i := \theta_{X_i}\omega\in\Omega, \qquad i\ge 0,
\end{equation}
is also a Markov chain, called the {\it environment viewed from the particle} process. By abuse of notation, we enlarge our probability space so that $P_\omega$ still denotes the joint law of the random walks and $(\bar\omega^i)_{i\ge 0}$.

For $\omega\in\Omega$ and $i\in \{1,\ldots,d\}$, we set $a_i(x)=a_i^\omega(x) := \frac{\omega_i(x)}{\tr\omega(x)} = 2\omega(x,x+e_i)$ and
\begin{equation}
\label{eq:def-a}
a(x)=a^\omega(x):=\diag[a_1(x),\ldots, a_d(x)]=\frac{\omega(x)}{\tr\omega(x)}, \quad x\in\Z^d.
\end{equation}
Finally, for $\omega\in \Omega$, we introduce the difference operator $L_{\omega}$ given by
\begin{align*}
  L_\omega u(x) := \sum_{y\in \Z^d}\omega(x,y)[u(y)-u(x)]=\tfrac{1}{2}\tr(a(x)\nabla^2 u(x)).
\end{align*}
That is, $L_\omega$ is the generator of the process $(Y_t)_{t\ge 0}$.

\subsection{Main assumptions and some notations}
Throughout the paper, the following assumptions are always in force:
\begin{enumerate}[(\text{A}1)]
\item $\left\{\omega(x),\, x\in \Z^d \right\}$ are i.i.d. under the probability measure $\mb P$.
\item\label{item:ellip-cond} $\frac{\omega}{\tr\omega}\ge 2\kappa{\rm I}$ for $\mb P$-almost every $\omega$ and some constant $\kappa\in(0,\tfrac{1}{2d}]$.
\item $\psi(\omega):=\tfrac{\zeta(\omega)}{\tr\omega(0)}$ is a $L^\infty(\mb P)$-bounded measurable function of $\omega(0)$.
\end{enumerate}
Throughout this paper, we use $c, C$ to denote positive constants which may change from line to line but only depend on the dimension $d$ and the ellipticity constant $\kappa$ unless stated otherwise. We write $A
\lesssim B$ if $A\le CB$, and $A\asymp B$ if $A\lesssim B$ and $A\gtrsim B$.
We also use the notations $A\lesssim_j B$ and $A\asymp_j B$ to indicate that the multiplicative constant depends on the variable $j$ other than $(d,\kappa)$.
We will use the convention of summation over repeated indices.
For example, $a_kb_k$ represents $\sum_{k=1}^d a_kb_k$.
This convention is not used in definitions \eqref{eq:def-lambda} and \eqref{eq:def-eta}.
For simplicity, we use $\tx=\tx(\omega)$ to denote a generic {\it stretched-exponentially integrable} random variable (i.e., $\mb E[\exp(\tx^c)]<C$) whose value may differ from line to line.
We write $\tx_x(\omega):=\tx(\theta_x\omega)$.
We note that $\tx$ may depend on other parameters as well, although the constants $c, C$ in the stretched-exponential moment bound only depend on $(d,\kappa)$.

\subsection{Earlier results in the literature}

We first recall the following quenched central limit theorem (QCLT) proved in \cite{L-82}, which is a discrete version of \cite{PV-82}.
\begin{thmx}\label{thm:QCLT}
Assume \rm{(A2)} and that law $\mb P$ of the environment is ergodic under spatial shifts $\{\theta_x:x\in\Z^d\}$. Then, the following assertions hold.
\begin{enumerate}[(i)]
\item\label{item:ergodic} There exists a probability measure $\mb Q\approx\mb P$ such that $(\evp{i})_{i\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\mb Q\times P_\omega$.
\item For $\mb P$-almost every $\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\omega$) to a Brownian motion with covariance matrix
\begin{equation}
\label{eq:def-abar}
\bar a=\diag[\bar a_1,\ldots,\bar a_d]:=E_\Q[a]=E_\Q[\tfrac{\omega(0)}{\tr\omega(0)}]>0.
\end{equation}
\end{enumerate}
\end{thmx}
We denote the Radon--Nikodym derivative of $\Q$ with respect to $\mb P$ by
\begin{equation}\label{eq:def-rho}
\rho(\omega)=\dd\mb Q/\dd\mb P.
\end{equation}
If the environment is i.i.d., it was proved in \cite[Theorem 1.5]{GT-22} that
\begin{equation}
    \label{eq:rho-bounds}
    E[\exp(\rho^{-c})]+E[\exp(\rho^{c})]<\infty
\end{equation}
for some constant $c=c(\kappa,d)>0$.
For any $x\in\Z^d$, we write
\[
\rho_\omega(x):=\rho(\theta_x\omega).
\]
For any $\alpha\in(0,1]$ and any function $f$ on a set $A$, we define
\[
\osc_A f:=\sup_{x,y\in A}|f(x)-f(y)|,
\qquad
[f]_{\alpha;A} :=\sup_{x,y\in A,\; x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha},
\]
and, if $A$ is a finite set, for $p\in(0,\infty)$, we define
\[
\norm{f}_{p;A}:=\left(\frac{1}{\#A}\sum_{x\in A}|f(x)|^p\right)^{1/p}, \quad
\norm{f}_{\infty;A} :=\max_{x\in A}|f(x)|.
\]

For any integer $j\ge 0$, let $\fct_j$ denote the set of $j$-th order polynomials, with $\fct_0=\R$. In fact, in our paper, we will only use the cases $j=0,1$.

The following large-scale regularity result can be found in \cite{GT-23}, which is a discrete version of the $C^{1,1}$ regularity estimate in \cite[Theorem 3.1]{AL-17}.
This was first done in the periodic setting in \cite{AL-87,AL-89}.

\begin{thmx}
\label{thm:c11}
Assume {\rm(A1), (A2)},  and that $\psi$ is a {\bf local} function. Let $R\ge 1$.
There exists $\alpha=\alpha(d,\kappa)\in(0,\tfrac13)$ such that,
for any $u$ with $L_\omega u(x)=\psi(\theta_x\omega)+f(x)$ on $B_R$, $j\in\{1,2\}$, $\tx\le r<R$, we have that
\begin{equation}
\label{eq:oscillation-estimates}
\frac{1}{r^{j}}\inf_{p\in\fct_{j-1}}\osc_{B_r}\,(u-p)
\lesssim
\frac{1}{R^{j}}
\inf_{p\in\fct_{j-1}}\osc_{B_R}\,(u-p)+A_{j},
\end{equation}
where the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds
\begin{align*}
&A_1\le R^{1-\alpha}\norm{\psi}_\infty+R\norm{f}_\infty
\quad\text{ and }\quad
A_1\le R^{1-\alpha}\norm{\psi+f(0)}_\infty+R^{1+\sigma}[f]_{\sigma;B_R},
\\
&A_2\le r^{-\alpha}\norm{\psi}_\infty+\log(\tfrac{R}{r})\norm{f}_\infty
\quad\text{ and }\quad
A_2\le r^{-\alpha}\norm{\psi+f(0)}_\infty+R^\sigma[f]_{\sigma;B_R}
\end{align*}
for any $\sigma\in(0,1]$. In particular, for any $R>1$ and $j\in\{1,2\}$, there holds
\begin{equation}\label{eq:c2}
|\nabla^ju(0)|\lesssim \left(\frac{\tx}{R}\right)^j
\left(
\norm{u-u(0)}_{1;B_R}+R^2\norm{\psi+f(0)}_\infty+R^{2+\sigma}[f]_{\sigma;B_{R}}
\right).
\end{equation}
\end{thmx}

Introducing the quantities
\begin{equation}\label{eq:def-mu}
\mu(R):=
\left\{
\begin{array}{lr}
R & d=2,\\
R^{1/2} &d=3,\\
(1\vee \log R)^{1/2} &d=4,\\
1 &d\ge 5,
\end{array}
\right.
\quad
\delta(R):=\left\{
\begin{array}{lr}
(\log (R\vee 2))^{3/2} &d=2,\\
1 &d\ge 3,
\end{array}
\right.
\end{equation}
the following theorems on properties of the (global) correctors and quantitative homogenization of the Dirichlet problem were established in \cite{GT-23}.

\begin{thmx}
\label{thm:global_krt}
Let $\psi$ be an $L^\infty(\mb P)$-bounded function of $\omega(0)$ with $\norm{\psi}_\infty=1$.
For each $d\ge 2$ and $\mb P$-a.e. $\omega$, there exists a function $\krt=\krt_\omega:\Z^d\to\R$ that solves
\[
L_\omega\krt(x)=\psi(\theta_x\omega)-\bar\psi \quad\text{ for }x\in\Z^d
\]
with the following properties:
\begin{enumerate}[(i)]
\item\label{item:gkrt-2} When $d\ge 5$,\quad $\mb E[\exp(c|\krt(x)/\mu(|x|)|^p)]<C_p$ for any $p\in(0,\tfrac{2d}{3d+2}),x\in\Z^d$;

When $d=3,4$,\; $\mb E[\exp(c|\krt(x)/\mu(|x|)|^p)]<C_p$ for any $p\in(0,\tfrac{2d}{3d+4}),x\in\Z^d$;

When $d=2$,\quad
$\mb E\left[
\exp\big(
c\abs{\tfrac{\krt(x)}{|x|\delta(|x|)}}^p
\big)
\right]\le C_p$ for any $p\in(0,\tfrac13), x\in\Z^d$;

\item\label{item:gkrt-3} $\mb E[\exp(c|\nabla\krt(x)/\delta(|x|)|^q)]\le C_q$ for any $q\in(0,\tfrac{d}{2d+2})$ and $x\in\Z^d$;
\item\label{item:gkrt-2nd-der}
 $\mb E[\exp(c|\nabla^2\krt(x)|^r)]\le C_r$ for any $r\in(0,\tfrac{d}{2})$ and $x\in\Z^d$;
\item\label{item:gkrt-4}(Stationarity properties)
\begin{itemize}
\item When $d\ge 5$,  the field $\{\krt_\omega(x):x\in\Z^d\}$ is stationary (under $\mb P$);
\item When $d\ge 3$, we have $\nabla\krt_\omega(x)=\nabla\krt_{\theta_x\omega}(0)$ for all $x\in\Z^d$, and the field $\{\nabla\krt_\omega(x):x\in\Z^d\}$  is stationary;
\item When $d=2$, we have $\nabla^2\krt_\omega(x)=\nabla^2\krt_{\theta_x\omega}(0)$ for all $x\in\Z^d$, and the field $\{\nabla^2\krt_\omega(x):x\in\Z^d\}$  is stationary.
\end{itemize}
\end{enumerate}
Moreover,  such a corrector $\krt$ is unique up to an additive constant when $d\ge 3$,  and is unique up to an affine transformation when $d=2$.
\end{thmx}

\begin{thmx}
\label{thm:opt-quant-homo}
Assume {\rm(A1)}, {\rm(A2)}, and {\rm(A3)}. Let $\bar a :=E_{\Q}[\frac{\omega(0)}{\tr\omega(0)}]>0$ and $\bar\psi :=E_{\Q}[\frac{\zeta}{\tr\omega(0)}]$. Suppose $f,g$ are both in $C^4(\R^d)$. Then, for any $\error\in(0,1)$ and $R\ge 2$, there exists a random variable $\ms Y=\ms Y(R,\error, \omega)>1$ with $\mb E[\exp(\ms Y^{{d}/{(2d+2)}-\error})]<C$ such that the solution $u$ of \eqref{eq:elliptic-dirich} satisfies
\[
\max_{x\in B_R}|u(x)-\bar{u}(\tfrac{x}{R})|
\lesssim
\left\{
\begin{array}{rl}
(\log R)^{2+\error}R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d=2,\\
R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d\ge 3,
\end{array}
\right.
\]
where $\bar u$ is the solution of the Dirichlet problem \eqref{eq:effective-ellip}.
\end{thmx}

We point out that Theorem \ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\  (for example, the ``checkerboard" environment considered in \cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \cite{GT-22} and the optimal convergence rates \cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.
We refer the reader to \cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.

\subsection{Main results}

For $x\in\Z^d$, a {\it spatial shift} is defined by
 \[
 \theta_x:\Omega\to\Omega,\qquad \theta_x\omega := \omega(x+\cdot).
 \]
In a random environment $\omega\in\Omega$ in dimensions $d\ge 2$, we consider the discrete {\it non-divergence form} elliptic Dirichlet problem
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
u(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
where $f,g\in\R^{\bar\B_1}$ are sufficiently regular, and $\zeta\in \R^\Omega$ satisfies suitable integrability conditions.
As $R\to\infty$, it is expected that $\vert u - \bar u(\frac{\cdot}{R})\rvert$ converges to $0$, where $\bar u$ solves
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
Here, $D^2 \bar{u}$ denotes the Hessian matrix of $\bar{u}$, and $\bar a=\bar a(\mb P)\in\mb S_{d\times d}$ and $\bar\psi=\bar\psi(\mb P,\zeta)\in\R$ are {\it deterministic} and do not depend on the realization of the random environment; see Theorem~\ref{thm:QCLT} and Theorem~\ref{thm:opt-quant-homo} for formulas of $\bar{a}$ and $\bar{\psi}$.

We first recall the following quenched central limit theorem (QCLT) proved in \cite{L-82}, which is a discrete version of \cite{PV-82}.
\begin{thmx}\label{thm:QCLT}
Assume \rm{(A2)} and that law $\mb P$ of the environment is ergodic under spatial shifts $\{\theta_x:x\in\Z^d\}$. Then, the following assertions hold.
\begin{enumerate}[(i)]
\item\label{item:ergodic} There exists a probability measure $\mb Q\approx\mb P$ such that $(\evp{i})_{i\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\mb Q\times P_\omega$.
\item For $\mb P$-almost every $\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\omega$) to a Brownian motion with covariance matrix
\begin{equation}
\label{eq:def-abar}
\bar a=\diag[\bar a_1,\ldots,\bar a_d]:=E_\Q[a]=E_\Q[\tfrac{\omega(0)}{\tr\omega(0)}]>0.
\end{equation}
\end{enumerate}
\end{thmx}
We denote the Radon--Nikodym derivative of $\Q$ with respect to $\mb P$ by
\begin{equation}\label{eq:def-rho}
\rho(\omega)=\dd\mb Q/\dd\mb P.
\end{equation}
If the environment is i.i.d., it was proved in \cite[Theorem 1.5]{GT-22} that
\begin{equation}
 \label{eq:rho-bounds}
 E[\exp(\rho^{-c})]+E[\exp(\rho^{c})]<\infty
\end{equation}
for some constant $c=c(\kappa,d)>0$.
For any $x\in\Z^d$, we write
\[
\rho_\omega(x):=\rho(\theta_x\omega).
\]
For any $\alpha\in(0,1]$ and any function $f$ on a set $A$, we define
\[
\osc_A f:=\sup_{x,y\in A}|f(x)-f(y)|,
\qquad
[f]_{\alpha;A} :=\sup_{x,y\in A,\; x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha},
\]
and, if $A$ is a finite set, for $p\in(0,\infty)$, we define
\[
\norm{f}_{p;A}:=\left(\frac{1}{\#A}\sum_{x\in A}|f(x)|^p\right)^{1/p}, \quad
\norm{f}_{\infty;A} :=\max_{x\in A}|f(x)|.
\]

\begin{thmx}
\label{thm:c11}
Assume {\rm(A1), (A2)}, and that $\psi$ is a {\bf local} function. Let $R\ge 1$.
There exists $\alpha=\alpha(d,\kappa)\in(0,\tfrac13)$ such that,
for any $u$ with $L_\omega u(x)=\psi(\theta_x\omega)+f(x)$ on $B_R$, $j\in\{1,2\}$, $\tx\le r<R$, we have that
\begin{equation}
\label{eq:oscillation-estimates}
\frac{1}{r^{j}}\inf_{p\in\fct_{j-1}}\osc_{B_r}\,(u-p)
\lesssim
\frac{1}{R^{j}}
\inf_{p\in\fct_{j-1}}\osc_{B_R}\,(u-p)+A_{j},
\end{equation}
where the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds
\begin{align*}
&A_1\le R^{1-\alpha}\norm{\psi}_\infty+R\norm{f}_\infty
\quad\text{ and }\quad
A_1\le R^{1-\alpha}\norm{\psi+f(0)}_\infty+R^{1+\sigma}[f]_{\sigma;B_R},
\\
&A_2\le r^{-\alpha}\norm{\psi}_\infty+\log(\tfrac{R}{r})\norm{f}_\infty
\quad\text{ and }\quad
A_2\le r^{-\alpha}\norm{\psi+f(0)}_\infty+R^\sigma[f]_{\sigma;B_R}
\end{align*}
for any $\sigma\in(0,1]$. In particular, for any $R>1$ and $j\in\{1,2\}$, there holds
\begin{equation}\label{eq:c2}
|\nabla^ju(0)|\lesssim \left(\frac{\tx}{R}\right)^j
\left(
\norm{u-u(0)}_{1;B_R}+R^2\norm{\psi+f(0)}_\infty+R^{2+\sigma}[f]_{\sigma;B_{R}}
\right).
\end{equation}
\end{thmx}

\begin{thmx}
\label{thm:opt-quant-homo}
Assume {\rm(A1)}, {\rm(A2)}, and {\rm(A3)}. Let $\bar a :=E_{\Q}[\frac{\omega(0)}{\tr\omega(0)}]>0$ and $\bar\psi :=E_{\Q}[\frac{\zeta}{\tr\omega(0)}]$. Suppose $f,g$ are both in $C^4(\R^d)$. Then, for any $\error\in(0,1)$ and $R\ge 2$, there exists a random variable $\ms Y=\ms Y(R,\error, \omega)>1$ with $\mb E[\exp(\ms Y^{{d}/{(2d+2)}-\error})]<C$ such that the solution $u$ of \eqref{eq:elliptic-dirich} satisfies
\[
\max_{x\in B_R}|u(x)-\bar{u}(\tfrac{x}{R})|
\lesssim
\left\{
\begin{array}{rl}
(\log R)^{2+\error}R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d=2,\\
R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d\ge 3,
\end{array}
\right.
\]
where $\bar u$ is the solution of the Dirichlet problem \eqref{eq:effective-ellip}.
\end{thmx}

We point out that Theorem \ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\ (for example, the ``checkerboard" environment considered in \cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \cite{GT-22} and the optimal convergence rates \cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.
We refer the reader to \cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.

\subsection{Main results}

\subsection{Main results}

\begin{proposition}
\label{lem:two-scale-exp}
Suppose that $d\geq 3$, $f\in C^{4,\alpha}(\bar\B_1)$, and $g\in C^{6,\alpha}(\partial\B_1)$. Let $u,\bar u$ be the solutions of \eqref{eq:elliptic-dirich}, \eqref{eq:effective-ellip}, respectively. Recall $a,\bar a$ from \eqref{eq:def-a}, \eqref{eq:def-abar}, and let $v^k,\xi,\lambda^k_j, \bar\lambda^k_j,\eta_j,\bar\eta_j$ be as above. We extend $g$ so that $g=\bar{u}$ on $\bar\B_1$. Let $\bar z:\bar\B_1\to\R$ be the solution of
\[
\left\{
\begin{array}{lr}
\frac{1}{2}\tr(\bar{a}D^2 \bar{z}(x))=\tfrac{1}{2}\bar{\lambda}^k_i\partial_{kki}\bar{u}(x)-\tfrac{1}{2}\bar\eta_j\partial_j f(x) & x\in\B_1,\\
\bar z(x)=0 &x\in\partial\B_1.
\end{array}
\right.
\]
For $\omega\in\Omega$, $R>1$, and $j,k\in\{1,\ldots,d\}$, let $p^k_j, s_j:\Z^d \rightarrow \R$ be solutions of
\begin{align}
L_\omega p^k_j(x)&=\lambda^k_j(x)-\bar\lambda^k_j \quad\text{ for }x\in B_R,\label{eq:def-p}\\
L_\omega s_j(x)&=\eta_j(x)-\bar\eta_j \quad\;\text{ for }x\in B_R\label{eq:def-sj},
\end{align}
respectively. Then, for $\omega\in\Omega$, $R>1$, we have
\begin{align}\label{eq:two-scale-further}
&\max_{x\in B_R}\left|u(x)-\bar{u}(\tfrac xR)-\tfrac 1R\bar z(\tfrac xR)\right|\nn\\
&\qquad\lesssim
\tfrac 1{R^2}\norm{\bar u}_{C^6(\bar{\B}_1)}\max_{i,j,k=1}^d
\bigg[1+\left\| |\nabla p^k_j|+|\nabla s_j|+(|\bar\lambda_j^i|+|\bar\eta_j|)|\nabla v^k|\right\|_{d;B_R}
\nn\\
&\qquad
\qquad\qquad\qquad\qquad\quad\;\;\,+\max_{\bar B_R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+|s_i|)\right)\bigg].
\end{align}
\end{proposition}

\begin{proof}
Let $z:\binB R\to\R$ be a solution of the Dirichlet problem
\[
\left\{
\begin{array}{lr}
L_\omega z(x)=\tfrac{1}{R^2}\left[ \tfrac{1}{2}\bar{\lambda}^k_i\partial_{kki}\bar{u}(\tfrac xR)-\tfrac{1}{2}\bar\eta_i\partial_i f(\tfrac xR)\right] & x\in\inB R,\\
z(x)=0 &x\in\partial\inB R.
\end{array}
\right.
\]
Consider the function
\begin{align}\label{eq:def-w}
&w(x)=u(x)-\bar u(\tfrac{x}{R})-\tfrac 1R z(x)\nn\\
&
+\tfrac{1}{R^2}\left[v^k(x)\partial_{kk}\bar u(\tfrac{x}{R})
-f(\tfrac{x}R)\xi(x)\right]- {\tfrac{1}{2}} \tfrac{1}{R^3}\left[\partial_{kki}\bar{u}(\tfrac{x}{R})p_i^k(x)-\partial_i f(\tfrac{x}{R})s_i(x)\right]
\end{align}
for $x\in\binB R\subset\bar\B_R\cap\Z^d$. Then, applying the formula
\begin{align*}
L_\omega(uv)(x)&=u(x)L_\omega v(x)+v(x) L_\omega u(x)+
\sum_{y:y\sim x}\omega(x,y)[u(y)-u(x)][v(y)-v(x)]\\
&=u(x)L_\omega v(x)+v(x) L_\omega u(x) \\ &\qquad+ \tfrac{1}{2}
a_i(x)\left(\nabla_{e_i}u(x)[v(x+e_i)-v(x-e_i)]
+\nabla_i^2u(x)\nabla_{-e_i}v(x)
\right)
\end{align*}
to the products within \eqref{eq:def-w}, we obtain in $\inB R$ that
\begin{align}
&L_\omega[v^k\,\partial_{kk}\bar u(\tfrac{\cdot}{R})]\nn\\
&=
L_\omega v^k \,\partial_{kk}\bar u(\tfrac{\cdot}{R})
+v^k \, L_\omega[\partial_{kk}\bar u(\tfrac{\cdot}{R})]+\tfrac{1}{2}\lambda^k_i\nabla_{e_i}[\partial_{kk}\bar u(\tfrac{\cdot}{R})]+\tfrac{1}{2}a_i \,\nabla_i^2[\partial_{kk}\bar u(\tfrac{\cdot}{R})]\,\nabla_{-e_i}v^k
\nn\\
&=\tfrac1{2}(a_k -\bar a_k)\partial_{kk}\bar u(\tfrac{\cdot}{R})+\tfrac{1}{2}\tfrac{1}{R}\lambda^k_i\partial_{kki}\bar u(\tfrac{\cdot}{R})+O(\tfrac 1{R^2})\norm{\bar u}_{C^5(\bar\B_1)}\max_{\binB R}\sum_{k = 1}^d|v^k|,\label{eq:term1}
\end{align}
where we used the fact that $|\lambda^k_i|\lesssim|\nabla v^k|\lesssim\max_{\binB R}|v^k|$ in $\inB R$. Similarly,
\begin{align}
L_\omega[f(\tfrac{\cdot}R)\xi ]
&=(\psi-\bar\psi)f(\tfrac{\cdot}R)+{\tfrac{1}{2}}\tfrac{1}{R}\eta_i\partial_i f(\tfrac{\cdot}{R})+O(\tfrac 1{R^2})\norm{\bar u}_{C^5(\bar\B_1)}\max_{\inB R}|\xi|,\label{eq:term2}\\
L_\omega[\partial_{kki}\bar{u}(\tfrac{\cdot}{R})p_i^k ]
&=(\lambda^k_i-\bar\lambda^k_i)\partial_{kki}\bar{u}(\tfrac{\cdot}{R})
+{\tfrac{1}{2}}\tfrac{1}{R}\partial_{kkij}\bar{u}(\tfrac{\cdot}{R})a_j[p^k_i(\cdot+e_j)-p^k_i(\cdot-e_j)]\nn\\
&+O(\tfrac 1{R^2})\norm{\bar u}_{C^6(\bar\B_1)}\max_{\binB R}\sum_{i,k = 1}^d|p^k_i|,\label{eq:term3}\\
L_\omega[\partial_i f(\tfrac{\cdot}{R})s_i ]
&=(\eta_i-\bar\eta_i)\partial_i f(\tfrac{\cdot}{R})
+{\tfrac{1}{2}}\tfrac{1}{R}\partial_{ij}f(\tfrac{\cdot}{R})a_j [s_i(\cdot+e_j)-s_i(\cdot-e_j)]\nn\\
&+O(\tfrac 1{R^2})\norm{\bar u}_{C^6(\bar\B_1)}\max_{\binB R}\sum_{i=1}^d|s_i|.\label{eq:term4}
\end{align}
Note that
$\abs{L_\omega[\bar u(\tfrac{\cdot}{R})]-\tfrac1{2R^2}\tr[a D^2\bar{u}(\tfrac{\cdot}{R})]}\lesssim\tfrac1{R^4}\norm{\bar u}_{C^4(\bar\B_1)}$, and
\begin{align*}
L_\omega u
=\tfrac1{R^2}f(\tfrac{\cdot}{R})(\psi -\bar\psi)+\tfrac1{R^2}f(\tfrac{\cdot}{R})\bar\psi =\tfrac1{R^2}\big[f(\tfrac{\cdot}{R})L_\omega\xi +
\tfrac1{2}\bar{a}_k\partial_{kk}\bar u(\tfrac{\cdot}{R})\big].
\end{align*}
Then, recalling the definition of $w$ in \eqref{eq:def-w} and writing
\[
A:=\max_{\binB R}\sum_{i,k=1}^d\left[|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+|s_i|)\right],
\]
by \eqref{eq:term1}, \eqref{eq:term2},\eqref{eq:term3}, \eqref{eq:term4},
we have in $\inB R$ that
\begin{align*}
&\abs{L_\omega w } \\
&=
\tfrac1{R^2}\Abs{
f(\tfrac{\cdot}{R})L_\omega\xi +\tfrac1{2}\bar{a}_k\partial_{kk}\bar u(\tfrac{\cdot}{R})-R^2L_\omega[\bar{u}(\tfrac{\cdot}{R})]-{\tfrac{1}{2}}\tfrac{1}{R} \bar{\lambda}^k_i\partial_{kki}\bar{u}(\tfrac{\cdot}{R})+{\tfrac{1}{2}}\tfrac 1R\bar\eta_i\partial_i f(\tfrac{\cdot}{R})
\\
&\qquad+\tfrac1{2}(a_k-\bar a_k)\partial_{kk}\bar u(\tfrac{\cdot}{R})+{\tfrac{1}{2}}\tfrac{1}{R}\lambda^k_i\partial_{kki}\bar u(\tfrac{\cdot}{R})
\nn\\
&\qquad-(\psi-\bar\psi)f(\tfrac{\cdot}{R})-{\tfrac{1}{2}}\tfrac{1}{R}\eta_i\partial_i f(\tfrac{\cdot}{R})+O(\tfrac{1}{R^2})\norm{\bar u}_{C^6(\bar\B_1)}A\\
&\qquad-{\tfrac{1}{2}}\tfrac{1}{R}(\lambda^k_i-\bar\lambda^k_i)\partial_{kki}\bar{u}(\tfrac{\cdot}{R})
-{\tfrac{1}{4}}\tfrac{1}{R^2}\partial_{kkij}\bar{u}(\tfrac{\cdot}{R})a_j[p^k_i(\cdot+e_j)-p^k_i(\cdot-e_j)]\\
&\qquad+{\tfrac{1}{2}}\tfrac{1}{R}(\eta_i-\bar\eta_i)\partial_i f(\tfrac{\cdot}{R})
+{\tfrac{1}{4}}\tfrac{1}{R^2}\partial_{ij}f(\tfrac{\cdot}{R})a_j[s_i(\cdot+e_j)-s_i(\cdot-e_j)]
}
\nn\\
&\lesssim
\tfrac1{R^4}\norm{\bar u}_{C^6(\bar\B_1)}\sum_{i,k=1}^d\left[1+
|\nabla p_i^k|+|\nabla s_i|
+\max_{\binB R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+ |s_i|)\right)
\right].
\end{align*}
Thus, by the above inequality, the definition of $w$ from \eqref{eq:def-w}, and the ABP maximum principle, we obtain that
\begin{align}
&\max_{\binB R}|u -\bar u(\tfrac{\cdot}{R})-\tfrac 1R z |\label{eq:ineq1}\\
&\le \max_{\binB R}|w|+
\max_{\binB R}\Abs{\tfrac{1}{R^2}[v^k\partial_{kk}\bar u(\tfrac{\cdot}{R})
-f(\tfrac{\cdot}{R})\xi ]
-{\tfrac{1}{2}}\tfrac{1}{R^3}[\partial_{kki}\bar{u}(\tfrac{\cdot}{R})p_i^k -\partial_i f(\tfrac{\cdot}{R})s_i ]}\nn\\
&\lesssim
R^2\norm{L_\omega w}_{d;\inB R}
+\tfrac1{R^2}\norm{\bar u}_{C^3}\max_{\binB R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+ |s_i|)\right)\nn
\\&
\lesssim
\tfrac1{R^2}\norm{\bar u}_{C^6}\sum_{i,k=1}^d
\big[1+
\left\||\nabla p_i^k|+|\nabla s_i|\right\|_{d;B_R}+\max_{\binB R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+|s_i|)\right)
\big].\nn
\end{align}
Furthermore, by \cite[Lemma~30]{GT-23},
\[
\max_{\binB R}\abs{z-\bar z(\tfrac{\cdot}{R})}
\lesssim
\tfrac1{R^2}\max_{j,\ell=1}^d(|\bar\lambda_j^\ell|+|\bar\eta_j|)\norm{\bar u}_{C^5(\bar\B_1)}
\sum_{k=1}^d(1+R \norm{\nabla v^k}_{d;\inB R}+\osc_{\binB R}v^k).
\]
This inequality, together with \eqref{eq:ineq1}, yields the claimed bound \eqref{eq:two-scale-further}.
\end{proof}

In the i.i.d. setting with well-prepared boundary conditions, Theorem \ref{thm:opt-iid-homo} yields improved convergence rates compared to the rate $O(R^{-1})$ in Theorem \ref{thm:opt-quant-homo} when $d\geq 3$.
Our approaches and contributions are as follows.
We employ the two-scale asymptotic expansion to obtain the estimate \eqref{eq:two-scale-further} in Proposition \ref{lem:two-scale-exp}.
Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\omega(\cdot)$ and $\omega(-\,\cdot)$ are identically distributed.
(Of course, this symmetry fails for environments with only a finite range of dependence.)
This reflection symmetry implies that the third-order homogenized tensor vanishes, which in turn suggests that a homogenization error bound sharper than
 $O(R^{-1})$ may be attainable.

We remark that such improved rates should not be expected in environments with correlations even slightly weaker than in the point-wise i.i.d.\ case, since the distribution's reflective symmetry is no longer present.

In our i.i.d.\ setting, without the presence of the third-order tensor, the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.
Roughly speaking, the (first-order) correctors “correct” local coefficients, while the higher-order correctors account for the derivatives of the lower-order correctors.
Hence, unlike the correctors themselves, the source terms in the equations for higher-order correctors are highly nonlocal, and thus the large-scale regularity estimates in Theorem~\ref{thm:c11}, which apply only to equations with {\it local} source terms, no longer hold.

Our improved convergence rates in Theorem \ref{thm:opt-iid-homo} are inspired by earlier results in the periodic setting \cite{GST-22}, where it was proved that if the environment has ``one degree of freedom" $a(x)$, i.e., the coefficient matrix is of the form $\omega(x)=C+a(x)M$ for some constant symmetric matrices $C$ and $M$, then the third-order homogenized tensor vanishes, leading to an improved rate of $O(R^{-2})$.
The periodic setting, however, is quite rigid.
We provided examples in \cite{GST-22} that show that if the environment has two degrees of freedom, then the third-order homogenized tensor does not vanish in general, and the optimal convergence rate in such cases is only $O(R^{-1})$.

In contrast, in our random setting, $\omega(x)=\mathrm{diag}[\omega_1(x),\ldots, \omega_d(x)]$ has $d$ degrees of freedom, which is fundamentally different from the periodic case considered in \cite{GST-22}.
Moreover, in the periodic setting, the $C^{0,1}$-boundedness of higher-order correctors is automatically guaranteed by periodicity.

One of the main contributions of our work is to establish $C^{0,1}$ bounds for higher-order correctors in the random setting. Our approach is based on comparing the true higher-order correctors with certain {\it localized} higher-order correctors that possess good regularity properties and {\it approximate} higher-order correctors that are stationary.

\begin{remark}
	When $d=2$, the corrector $\upsilon^k$ grows super-linearly (Theorem~\ref{thm:global_krt}). Consequently, via the expansion \eqref{eq:two-scale-further}, it produces an error term of order $R^{-2}|v^k|$ that dominates the desired $O(R^{-1})$ scale. Therefore, in two dimensions, we should not expect an improvement beyond the $O(R^{-1})$ rate in i.i.d.\ environments.

    All in all, this demonstrates that, in the absence of additional structural assumptions on the random environment, the boundary conditions, or the dimension, the best convergence rate that one can expect is $O(R^{-1})$, as established in \cite{GT-23}. We refer to Remark~\ref{rmk:optimality} for a discussion of the optimality of the $O(R^{-1})$ rate.
\end{remark}

To prove Theorem~\ref{thm:opt-iid-homo}, the authors describe the following strategy and key ingredients.

- They “employ the two-scale asymptotic expansion to obtain the estimate \eqref{eq:two-scale-further} in Proposition~\ref{lem:two-scale-exp}.”

- They use a special feature of the point-wise i.i.d. setting: “Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\omega(\cdot)$ and $\omega(-\,\cdot)$ are identically distributed.” This “reflection symmetry implies that the third-order homogenized tensor vanishes,” which “suggests that a homogenization error bound sharper than $O(R^{-1})$ may be attainable.”

- With “the absence of the third-order tensor,” they explain that “the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.” They motivate the difficulty: while first-order correctors “correct local coefficients,” “the higher-order correctors account for the derivatives of the lower-order correctors,” and “the source terms in the equations for higher-order correctors are highly nonlocal,” so “the large-scale regularity estimates in Theorem~\ref{thm:c11}, which apply only to equations with {\it local} source terms, no longer hold.”

- A main technical step is then to “establish $C^{0,1}$ bounds for higher-order correctors in the random setting,” by “comparing the true higher-order correctors with certain {\it localized} higher-order correctors that possess good regularity properties and {\it approximate} higher-order correctors that are stationary.”

- They also note limitations: the improved rates rely on i.i.d.-driven reflection symmetry (“such improved rates should not be expected in environments with correlations… since the distribution's reflective symmetry is no longer present”), and in $d=2$ the corrector “grows super-linearly,” so the expansion yields an error term that “dominates the desired $O(R^{-1})$ scale,” hence “we should not expect an improvement beyond the $O(R^{-1})$ rate” in two dimensions.

\label{thm:opt-iid-homo}
Assume {\rm(A1), (A2), (A3)}, and $d\ge 3$. Let $\bar a,\bar\psi$ be as in the following theorem.

\begin{thmx}
\label{thm:opt-quant-homo}
Assume {\rm(A1)}, {\rm(A2)}, and {\rm(A3)}. Let $\bar a :=E_{\Q}[\frac{\omega(0)}{\tr\omega(0)}]>0$ and $\bar\psi :=E_{\Q}[\frac{\zeta}{\tr\omega(0)}]$. Suppose $f,g$ are both in $C^4(\R^d)$. Then, for any $\error\in(0,1)$ and $R\ge 2$, there exists a random variable $\ms Y=\ms Y(R,\error, \omega)>1$ with $\mb E[\exp(\ms Y^{{d}/{(2d+2)}-\error})]<C$ such that the solution $u$ of
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
\nu(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
satisfies
\[
\max_{x\in B_R}|u(x)-\bar{u}(\tfrac{x}{R})|
\lesssim
\left\{
\begin{array}{rl}
(\log R)^{2+\error}R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d=2,\\
R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d\ge 3,
\end{array}
\right.
\]
where $\bar u$ is the solution of the Dirichlet problem
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
\end{thmx}

Suppose that $f\in C^{4,\alpha}(\bar\B_1)$ and $g\in C^{6,\alpha}(\partial\B_1)$ for some $\alpha\in (0,1)$. Then, there exists a $C^6(\bar\B_1)$-extension of $g$ such that the solutions $u, \bar u$ of
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
\nu(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
and
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
satisfy
\[
\max_{x\in B_R}|u(x)-\bar u(\tfrac{x}{R})|
\lesssim
\left\{
  \begin{array}{rl}
  	R^{-3/2}\norm{\bar u}_{C^6(\bar\B_1)}\tx &\text{ when } d=3,\\
  	R^{-2}(\log R)\norm{\bar u}_{C^6(\bar\B_1)}\tx&\text{ when }d\ge 4.
  \end{array}
\right.
\]
An example of such an extension is $g=\bar u$.

Question:
Assume \((A1)\), \((A2)\), and \((A3)\), and let \(d\ge 3\). For a random environment \(\omega\), define the spatial shift by \(\theta_x\omega=\omega(x+\cdot)\). Let \(\bar a:=E_{\mathbb Q}[\omega(0)/\operatorname{tr}\omega(0)]>0\) and \(\bar\psi:=E_{\mathbb Q}[\zeta/\operatorname{tr}\omega(0)]\). Suppose \(f\in C^{4,\alpha}(\overline{\mathbb B}_1)\) and \(g\in C^{6,\alpha}(\partial\mathbb B_1)\) for some \(\alpha\in(0,1)\). Let \(u\) solve the discrete Dirichlet problem on the lattice ball \(B_R\subset\mathbb Z^d\),
\[
\tfrac12\operatorname{tr}(\omega(x)\nabla^2u(x))=\frac1{R^2}f\!\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega)\quad \text{for }x\in B_R,\qquad
u(x)=g\!\left(\tfrac{x}{R}\right)\quad \text{for }x\in \partial B_R,
\]
and let \(\bar u\) solve on the unit ball \(\mathbb B_1\subset\mathbb R^d\),
\[
\tfrac12\operatorname{tr}(\bar a\,D^2\bar u(x))=f(x)\bar\psi\quad \text{for }x\in\mathbb B_1,\qquad
\bar u(x)=g(x)\quad \text{for }x\in\partial\mathbb B_1.
\]
Which quantitative estimate holds for the homogenization error \(\max_{x\in B_R}|u(x)-\bar u(x/R)|\)?

There exists a \(C^6(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2}(\log R)\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]

Choices:
B.
There exists a \(C^6(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-2}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]
C.
There exists a \(C^6(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim R^{-1}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X,
\qquad d\ge 3.
\]
D.
For every \(\varepsilon\in(0,1)\), there exists a random variable \(\tilde X=\tilde X(R,\varepsilon,\omega)>1\) with stretched-exponential moments such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2+\varepsilon}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2+\varepsilon}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]
E.
There exists a \(C^4(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}(\log R)\,\|\bar u\|_{C^4(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2}\,\|\bar u\|_{C^4(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]

dimension-dependent rates and logarithmic factor

sharp improved rates dropped in favor of the coarser \(R^{-1}\) decay

required \(C^6\)-extension/norm and placement of logarithmic correction across dimensions

The stem does not reveal the correct option directly. It gives the setup and asks which existence statement is valid, without embedding the sharp rate or the key existential-vs-universal extension quantifier.

The item is close to a theorem-recall question: the correct choice is essentially the precise theorem statement with its rates and extension claim. However, the alternatives do introduce competing conclusions rather than merely asking for a verbatim restatement.

Selecting the correct answer requires some discrimination among subtle variants: sharp dimension-dependent rates, logarithmic loss placement, and the existential quantifier on the extension. Still, this is mostly recognition/recall of a known result rather than substantial generative mathematical reasoning.

The distractors are strong and plausible. They target natural failure modes: overclaiming a stronger rate, accepting a weaker but true bound, mishandling the quantifier over extensions, and misplacing the logarithmic factor across dimensions.

A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests precise theorem recall/discrimination rather than deeper generative reasoning.


\begin{theorem}\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\Omega}(t)$ has an expansion as $t\to 0$ given by
\begin{multline}\label{eq:heattrace}
H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\, ds + \sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)\\
+ \sqrt t\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big) +O(t\log t),
\end{multline}
where $\mathcal C_{1/2}$ is a function only of the angle $\alpha$ and the limit curvatures $\kappa_{\pm}$.
\end{theorem}

\begin{theorem}\label{thm:moreid} For $\alpha\ne\pi$, the function $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$ has the form
\begin{equation}\label{eq:form} \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-}) = c_{1/2}(\alpha)r_0(\alpha,\kappa_+,\kappa_-)
\end{equation}
where $c_{1/2}(\alpha)$ is a function depending only on $\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:
\begin{equation}\label{eq:general} \begin{aligned}
        c_{1/2}(\alpha)
        &= -\,\fp
        \iint_{(0,\infty)\times(0,\alpha)}
        \int_{0}^{1} (1-\sigma)^{-1}\sigma^{-1/2}
        \iint_{(0,\infty)\times(0,\alpha)} \\
        &\quad a\left(\frac{R}{\sqrt{1-\sigma}},\theta,
        \frac{\sqrt{\sigma}\,\rho}{\sqrt{1-\sigma}},\phi\right)\,
        b\left(\rho,\phi,\frac{R}{\sqrt{\sigma}},\theta\right)\,
        \dmup\,\dmu.
        \end{aligned}
\end{equation}
Here $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\alpha$, explicitly given by \eqref{eq:a-def} and \eqref{eq:bexpression-clean}, and $\fp$ denotes the Hadamard finite part at $R=\infty$ of the improper integral in $R$.

Moreover, $c_{1/2}(\alpha)$ has the special value
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
\end{theorem}

\begin{equation}\label{eq:a-def}
    a(R,\theta,R',\theta')
    =\frac12\,\exp \Big[-\frac{R^2+(R')^2}{4}\Big]
    \sum_{j=1}^\infty I_{\mu_j} \Big(\frac{RR'}{2}\Big)\,
    \varphi_j(\theta)\,\varphi_j(\theta').
\end{equation}

\begin{equation}\label{eq:bexpression-clean}
    \begin{aligned}
    b(R,\theta,R',\theta')&=
    \Big(\tfrac12 D^2S\Big)\,a(R,\theta,R',\theta')\\
    &\quad+\,4 R\cos(\theta+\theta_0)\,\Delta_{R,\theta}a(R,\theta,R',\theta')\\
    &\quad+\,\tfrac{1}{2}\,\Delta_{R,\theta}\big(D^2S\cdot a(R,\theta,R',\theta')\big)
    -\tfrac{1}{2}\,(D^2S)\,\Delta_{R,\theta}a(R,\theta,R',\theta').
    \end{aligned}
    \end{equation}

A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\mathbb R^2$ \cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \cite{Cheeger1983,vanDenBergSrisat88,NRS}.

We now explain the setting of this paper. Let $\Omega\subseteq\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\alpha_j>0$, $j=1,\ldots,n$. Let $\kappa$ be the inward-pointing curvature of the boundary, and let $\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\kappa_{j,\pm}$ be the limits of $\kappa(\gamma(s))$ as $s$ approaches $\gamma^{-1}(P_j)$ from above and below respectively.

It is well known that the trace of the Dirichlet heat kernel on $\Omega$, $H^D_{\Omega}(t)$, has an asymptotic expansion as $t\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \cite{Cheeger1983} and \cite{vdBS}, and then extended to the general curvilinear setting \cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \cite{EGS_2017}, based on results in the literature such as \cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.

It is well known that the trace of the Dirichlet heat kernel on $\Omega$, $H^D_{\Omega}(t)$, has an asymptotic expansion as $t\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \cite{Cheeger1983} and \cite{vdBS}, and then extended to the general curvilinear setting \cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \cite{EGS_2017}, based on results in the literature such as \cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.

Moreover, $c_{1/2}(\alpha)$ has the special value
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
\end{theorem}
\begin{remark} A few comments:
\begin{itemize}
\item The $O(t\log t)$ error in \eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.
\item Each of $a$ and $b$ is a function only of $\alpha$ and the variables of integration, see \eqref{eq:a-def}, \eqref{eq:bexpression-clean}. Thus \eqref{eq:general} is indeed independent of $\kappa_{\pm}$.
\item For inverse spectral results, it is crucial that \eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\alpha=\pi/2$. It would be of significant interest to find out for which $\alpha$ we have $c_{1/2}(\alpha)\ne 0$.
\item Theorem \ref{thm:moreid} does not work for $\alpha=\pi$ because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\alpha=\pi$. So the case $\alpha=\pi$, where the boundary has an abrupt change in second derivative, remains open.
\end{itemize}
\end{remark}

\begin{proposition}
 Suppose that an admissible curvilinear polygon $\Omega$ is Dirichlet isospectral to a polygon. Then $\Omega$ is a polygon.
\end{proposition}
\begin{proof}
For a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\Omega$. But under our assumptions, by Theorems \ref{thm:thm1} and \ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\Omega}(t)$ is given by
 \begin{equation}
 \frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n c_{1/2}(\alpha_j)\sqrt{\kappa_{+,j}^2+\kappa_{-,j}^2}.
 \end{equation}
 Since this coefficient is non-negative and only zero for an exact polygon, $\Omega$ is an exact polygon.
\end{proof}
\begin{remark}
Observe from \eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.
\end{remark}

\subsection{Plan of the paper}
In section \ref{sec:gm} we prove Theorem \ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \cite{NRS}. We show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$, and then prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$. To take advantage of this uniformity, in section \ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \cite{NRS} and allows us to obtain both \eqref{eq:form} and \eqref{eq:general}.
Finally, in section \ref{sec:specialpi2}, we prove \eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \eqref{eq:heattrace}.

\begin{lemma}\label{lem:local-corner}
Let $\Omega$ and $\Omega'$ be domains with a single corner having the same opening angle
$\alpha$ and the same one‑sided boundary curvatures $\kappa_\pm$ at that corner.
Then $H^D_\Omega-H^D_{\Omega'}$ has order $\le T^{0}$ (at worst $T^0 = t^0$) at ff.
Equivalently, their ff coefficients of orders $T^{-2} = t^{-1}$ and $T^{-1} = t^{-1/2}$ agree.
\end{lemma}

We can now prove Theorem \ref{thm:thm1}.
\begin{proof}[Proof of Theorem \ref{thm:thm1}] As in \cite{NRS}, the heat trace is obtained by restricting $H^{D}_{\Omega}$ to the spatial diagonal and then integrating over $z\in\Omega$. The technical tool used is Melrose's pushforward theorem. We will not repeat all the details here, but the upshot is that each face td, sf, and ff gives a separate contribution to the expansion \eqref{eq:heattrace}. A priori, the expansions can interact to give logarithmic terms, but for the same reasons as in \cite[p. 49]{NRS}, keeping in mind that we now have two orders at ff rather than one, there are no logarithmic terms until at least $O(T^2\log T)=O(t\log t)$ (and potentially much later, if at all). The contribution from td is
\[\frac{|\Omega|}{4\pi t} + O(t^{\infty}).\]
The contributions from sf are
\[-\frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\int_{\partial\Omega}\kappa\, ds +\frac{\sqrt t}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\, ds + O(t).\]
And the contributions from ff, since the first is already known and the second depends only on $\alpha$ and $\kappa_{\pm}$, are
\[\sum_{j=1}^{n}\frac{\pi^2-\alpha_j^2}{24\pi\alpha_j} + \sqrt t\sum_{j=1}^{n}\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-}) + O(t),\]
for some unknown function $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$. Summing the three faces gives Theorem \ref{thm:thm1}.
\end{proof}

For $H(t)$, we use the asymptotics of Bessel function zeroes due to McMahon \cite[(10.21.19)]{NIST:DLMF}:
\begin{equation}
j_{0,k}= (k-\frac 14)\pi + \frac{1}{8(k-\frac 14)\pi} + O(k^{-3}),
\end{equation}
which imply
\begin{equation}
j_{0,k}^{2} = (k-\frac 14)^2\pi^2 + \frac 14 + O(k^{-2}).
\end{equation}
Based on this we define the function
\begin{equation}
\tilde H(t) = \sum_{k=1}^{\infty} \exp[-t((k-\frac 14)^2\pi^2 + \frac 14)]
\end{equation}
and estimate its difference with $H(t)$.
\begin{lemma}\label{lem:difference} With all notation as above, as $t\to 0$,
\begin{equation}
H(t) - \tilde H(t) = O(t).
\end{equation}
\end{lemma}
This is helpful because we also have the following lemma.
\begin{lemma}\label{lem:compute} As $t\to 0$,
\begin{equation}
\tilde H(t) = \frac{1}{2\sqrt\pi}t^{-1/2} + c - \frac{1}{8\sqrt\pi}t^{1/2}+ O(t).
\end{equation}
Here $c$ is a constant which is irrelevant to our purposes.
\end{lemma}
From these two Lemmas and \eqref{eq:relationship} we immediately deduce this proposition.
\begin{proposition} As $t\to 0$,
\begin{equation}
\Tr e^{-t\Delta_{\Omega}} = \frac 18 t^{-1} - (\frac{\sqrt\pi}{8}+\frac{1}{4\sqrt\pi})t^{-1/2}-\frac 12c + (\frac{\sqrt\pi}{256}+\frac{1}{16\sqrt\pi})t^{1/2}+ O(t).
\end{equation}
\end{proposition}
\begin{remark} We could reverse engineer $c$ from the known expansion from \cite{NRS} but there is no need.
\end{remark}

We can say substantially more about $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$ whenever $\alpha\ne\pi$. Define
\begin{equation}\label{eq:defofr0}
    r_0=r_0(\alpha,\kappa_{+},\kappa_-)=\frac 12\csc\alpha\sqrt{\kappa_+^2+\kappa_-^2-2\kappa_+\kappa_-\cos\alpha}.
\end{equation}
\begin{theorem}\label{thm:moreid} For $\alpha\ne\pi$, the function $\mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-})$ has the form
\begin{equation}\label{eq:form} \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-}) = c_{1/2}(\alpha)r_0(\alpha,\kappa_+,\kappa_-)
\end{equation}
where $c_{1/2}(\alpha)$ is a function depending only on $\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:
\begin{equation}\label{eq:general} \begin{aligned}
        c_{1/2}(\alpha)
        &= -\,\operatorname*{fp}
        \iint_{(0,\infty)\times(0,\alpha)}
        \int_{0}^{1} (1-\sigma)^{-1}\sigma^{-1/2}
        \iint_{(0,\infty)\times(0,\alpha)} \\
        &\quad a\left(\frac{R}{\sqrt{1-\sigma}},\theta,
        \frac{\sqrt{\sigma}\,\rho}{\sqrt{1-\sigma}},\phi\right)\,
        b\left(\rho,\phi,\frac{R}{\sqrt{\sigma}},\theta\right)\,
        \,\rho\,d\rho\,d\phi\,\,R\,dR\,d\theta.
        \end{aligned}
\end{equation}
Here $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\alpha$, explicitly given by \eqref{eq:a-def} and \eqref{eq:bexpression-clean}, and $\operatorname*{fp}$ denotes the Hadamard finite part at $R=\infty$ of the improper integral in $R$.

Moreover, $c_{1/2}(\alpha)$ has the special value
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
\end{theorem}
\begin{remark} A few comments:
\begin{itemize}
\item The $O(t\log t)$ error in \eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.
\item Each of $a$ and $b$ is a function only of $\alpha$ and the variables of integration, see \eqref{eq:a-def}, \eqref{eq:bexpression-clean}. Thus \eqref{eq:general} is indeed independent of $\kappa_{\pm}$.
\item For inverse spectral results, it is crucial that \eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\alpha=\pi/2$. It would be of significant interest to find out for which $\alpha$ we have $c_{1/2}(\alpha)\ne 0$.
\item Theorem \ref{thm:moreid} does not work for $\alpha=\pi$  because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\alpha=\pi$. So the case $\alpha=\pi$, where the boundary has an abrupt change in second derivative, remains open.
\end{itemize}
\end{remark}

\subsection{Inverse spectral applications}

In \cite{EGS_2017}, the authors prove that any curvilinear polygon with straight corners is in fact isospectral to a polygon. They require straight corners because of a lack of a formula in the curved corner case. We can strengthen their result.
\begin{definition}
    A curvilinear polygon $\Omega$ is \emph{admissible} if, for each $j$, either $c_{1/2}(\alpha_j)>0$  or $\Omega$ is straight near $P_j$.
\end{definition}

\begin{proposition}
    Suppose that an admissible curvilinear polygon $\Omega$ is Dirichlet isospectral to a polygon. Then $\Omega$ is a polygon.
\end{proposition}
\begin{proof}
For a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\Omega$. But under our assumptions, by Theorems \ref{thm:thm1} and \ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\Omega}(t)$ is given by
    \begin{equation}
    \frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n c_{1/2}(\alpha_j)\sqrt{\kappa_{+,j}^2+\kappa_{-,j}^2}.
    \end{equation}
 Since this coefficient is non-negative and only zero for an exact polygon, $\Omega$ is an exact polygon.
\end{proof}
\begin{remark}
Observe from \eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.
\end{remark}

\subsection{Plan of the paper}
In section \ref{sec:gm} we prove Theorem \ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \cite{NRS}. We show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$, and then prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$. To take advantage of this uniformity, in section \ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \cite{NRS} and allows us to obtain both \eqref{eq:form} and \eqref{eq:general}.
Finally, in section \ref{sec:specialpi2}, we prove \eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \eqref{eq:heattrace}.

In the “Plan of the paper” the authors outline the proof of Theorem~\ref{thm:thm1}: in Section~\ref{sec:gm} they “examin[e] the geometric microlocal description of the heat kernel given in \cite{NRS},” “show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$,” and “prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$.” Then, “to take advantage of this uniformity,” in Section~\ref{sec:conformal} they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model,” via “a parametrix construction as in \cite{NRS},” which “allows [them] to obtain both \eqref{eq:form} and \eqref{eq:general}.” Finally, in Section~\ref{sec:specialpi2} they prove \eqref{eq:specialpi2} “by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \eqref{eq:heattrace}.”

In the “Plan of the paper” the authors outline the proof of the main theorem: next they “examin[e] the geometric microlocal description of the heat kernel given in NRS,” “show that there is a well-defined corner contribution $\mathcal C_{1/2}\sqrt t$,” and “prove that any two corners with the same angle $\alpha$ and limit curvatures $\kappa_{\pm}$ yield the same contribution, regardless of the rest of $\Omega$.” Then, “to take advantage of this uniformity,” they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\mathcal C_{1/2}\sqrt t$ in that model,” via “a parametrix construction as in NRS,” which “allows [them] to obtain both
\begin{equation}\label{eq:form} \mathcal C_{1/2}(\alpha,\kappa_{+},\kappa_{-}) = c_{1/2}(\alpha)r_0(\alpha,\kappa_+,\kappa_-)
\end{equation}
and
\begin{equation}\label{eq:general} \begin{aligned}
        c_{1/2}(\alpha)
        &= -\,\fp
        \iint_{(0,\infty)\times(0,\alpha)}
        \int_{0}^{1} (1-\sigma)^{-1}\sigma^{-1/2}
        \iint_{(0,\infty)\times(0,\alpha)} \\
        &\quad a\left(\frac{R}{\sqrt{1-\sigma}},\theta,
        \frac{\sqrt{\sigma}\,\rho}{\sqrt{1-\sigma}},\phi\right)\,\
        b\left(\rho,\phi,\frac{R}{\sqrt{\sigma}},\theta\right)\,\
        \dmup\,\dmu.
        \end{aligned}
\end{equation}.
Finally, they prove
\begin{equation}\label{eq:specialpi2} c_{1/2}(\frac{\pi}{2}) = \frac{1}{16\sqrt\pi}.
\end{equation}
“by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with
\begin{multline}\label{eq:heattrace}
H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\, ds + \sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)\\
+ \sqrt t\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big) +O(t\log t),
\end{multline}.”,

\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\Omega}(t)$ has an expansion as $t\to 0$ given by
\begin{multline}\label{eq:heattrace}
H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t} - \frac{|\partial\Omega|}{8\sqrt{\pi t}} + \frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\, ds + \sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)\\
+ \sqrt t\Big(\frac{1}{256\sqrt{\pi}}\int_{\partial\Omega}\kappa^2\, ds + \sum_{j=1}^n\mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big) +O(t\log t),
\end{multline}
where $\mathcal C_{1/2}$ is a function only of the angle $\alpha$ and the limit curvatures $\kappa_{\pm}..

Question: Let \(\Omega\subseteq\mathbb R^2\) be a curvilinear polygon: its boundary is piecewise smooth, smooth except at finitely many corners \(P_1,\dots,P_n\), where the interior angle at \(P_j\) is \(\alpha_j>0\). Let \(\kappa\) denote the inward-pointing curvature along the smooth boundary arcs, and for each corner let \(\kappa_{j,+}\) and \(\kappa_{j,-}\) be the one-sided limits of \(\kappa\) as the corner is approached along the two incident sides. Let \(H^D_{\Omega}(t)\) be the Dirichlet heat trace of \(\Omega\). As \(t\to 0\), which asymptotic expansion with error bound is valid?

\[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) depends only on the corner angle \(\alpha\) and the one-sided limiting curvatures \(\kappa_\pm\).

Choices:
B. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j)\sqrt{\kappa_{j,+}^2+\kappa_{j,-}^2}\Big)+O(t\log t),\]\nwhere \(\mathcal C_{1/2}(\alpha)\) depends only on the corner angle.
C. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+O(\sqrt t),\]\nas \(t\to 0\).
D. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big)+O(t),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) depends only on the corner angle \(\alpha\) and the one-sided limiting curvatures \(\kappa_\pm\).
E. \[H^D_{\Omega}(t)=\frac{|\Omega|}{4\pi t}-\frac{|\partial\Omega|}{8\sqrt{\pi t}}+\frac{1}{12\pi}\Big(\int_{\partial\Omega}\kappa\,ds+\sum_{j=1}^n\frac{\pi^2-\alpha_j^2}{2\alpha_j}\Big)+\sqrt t\Big(\frac{1}{256\sqrt\pi}\int_{\partial\Omega}\kappa^2\,ds+\sum_{j=1}^n \mathcal C_{1/2}(\alpha_j,\kappa_{j,+},\kappa_{j,-})\Big)+O(t^{3/2}),\]\nwhere \(\mathcal C_{1/2}(\alpha,\kappa_+,\kappa_-)\) may depend on the global geometry of \(\Omega\) in addition to \(\alpha\) and \(\kappa_\pm\).

corner-term dependence on full pair \((\kappa_+,\kappa_-)\) rather than a forced factorization through Euclidean norm

explicit \(\sqrt t\)-coefficient and sharper remainder \(O(t\log t)\)

error term improved from first guaranteed \(O(t\log t)\) to \(O(t)\)

uniform local determination of corner contribution by \(\alpha\) and \(\kappa_\pm\)

The stem introduces the geometric setting and asks for the valid asymptotic expansion, but it does not explicitly state the correct remainder term or the precise dependence of the corner contribution. There is no direct or trivial hint to the correct choice.

The item is close to a theorem-recall question: the correct option is essentially the full theorem statement. However, it is not a pure restatement, since the alternatives vary in subtle but meaningful ways (error bounds, locality of corner terms, and factorization claims).

Selecting the best answer requires moderate reasoning about which asymptotic claim is strongest validly, versus merely weaker true or falsely sharpened statements. Still, success depends largely on recognition/recall of the theorem rather than substantial derivation or synthesis.

The distractors are strong and mathematically plausible. They target realistic failure modes: replacing local dependence by an unjustified factorization, weakening to a less informative but true bound, and asserting overly strong remainders or improper global dependence. They are distinct and well aligned with expert-level misconceptions.

A solid high-level MCQ with no answer leakage and excellent distractors, but it is still primarily a near-verbatim theorem identification task rather than a deeply generative reasoning problem.