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Yinshan Chang

Publications and source records attributed to Yinshan Chang.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative Rigidity of pseudo-rotations on the two-torus and Sarnak's conjecture

We establish quantitative rigidity results for pseudo-rotations of the two-torus under a $(C,δ)$-deviation condition relative to their rotation vectors. The main ingredient is a quantitative free-disk estimate that converts bounds on orbit deviation into explicit control of the distance between the iterates and the identity map. Under such a $(C,δ)$-deviation condition, we show that Hölder continuous super-Liouvillean irrational pseudo-rotations are $C^0$-rigid with an exponential decay rate and that $C^k$ semi-irrational pseudo-rotations of strong non-Brjuno type exhibit $C^{k-1}$-rigidity with a superpolynomial decay rate. Moreover, under this deviation condition and sufficiently large irrationality measure, we show that Hölder continuous skew products on $\mathbb{T}^2$ over circle rotations are $C^0$-rigid with a polynomial decay rate. As a consequence, all these classes satisfy Sarnak's conjecture.

math.DS↗
arXiv2026

Regularity, quantitative deviation, and non-rigidity of a lacunary skew product

Let $α$ be irrational and let $q_j$ be the denominators of its continued-fraction convergents. We study the function \[ h(x)=\sum_{j\geq1}\frac{\cos(2πq_jx)}{q_j} \] and the skew product \[f(x,y)=(x+α,y+h(x))\quad\mathrm{mod}\quad\mathbb{Z}^2.\] The function $h$ is Hölder continuous of every exponent below one. A Fourier argument shows that $h$ is not Lipschitz. The map $f$ is a toral pseudo-rotation with rotation vector $(α,0)$, but it has neither bounded mean motion nor $C^0$-rigidity. Suppose $α$ satisfies the Diophantine condition $\mathrm{DC}(τ)$. Then, $f$ has $(C,1-1/τ)$-deviation when $τ>1$; and it has $(C_δ,δ)$-deviation for every $0<δ<1$, but not for $δ=0$ when $τ=1$.

math.DS↗
arXiv2026

Examples beyond Bounded Mean Motion for Quantitative Rigidity on the Two-Torus

This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<δ<\tfrac12$, we construct $C^\infty$ Lebesgue-area-preserving pseudo-rotations of $\mathbb{T}^2$ which satisfy the $(C,δ)$-deviation condition but do not have bounded mean motion. We give both semi-irrational and totally irrational rotation vectors and two realizations: a controlled weakly mixing Anosov--Katok construction and an explicit weakly mixing special flow construction. Weak mixing is used as a conjugacy-invariant obstruction to every continuous circle-rotation factor. Consequently, none of the resulting maps is topologically conjugate, by a linear or nonlinear change of coordinates, to a skew product over a circle rotation. In the special-flow realization, the same lacunary Fourier series simultaneously gives weak mixing, the sharp upper bound $O(n^δ)$, and unbounded deviations; in fact no smaller deviation exponent is possible. The semi-irrational examples meet the assumptions of Theorems~1 and~2 of the cited manuscript, whereas the totally irrational examples meet those of Theorem~1. Each construction produces continuum many maps and continuum many topological conjugacy classes of each rotation type.

math.DS↗
arXiv2026

On the maximal correlation of some stochastic processes

We study the maximal correlation coefficient $R(X,Y)$ between two stochastic processes $X$ and $Y$. In the case when $(X,Y)$ is a random walk, we find $R(X,Y)$ using the Csáki-Fischer identity and the lower semicontinuity of the map $\text{Law}(X,Y) \to R(X,Y)$. When $(X,Y)$ is a two-dimensional Lévy process, we express $R(X,Y)$ in terms of the Lévy measure of the process and the covariance matrix of the diffusion part of the process. Consequently, for a two-dimensional $α$-stable random vector $(X,Y)$ with $0<α<2$, we express $R(X,Y)$ in terms of $α$ and the spectral measure $τ$ of the $α$-stable distribution. We also establish analogs and extensions of the Dembo-Kagan-Shepp-Yu inequality and the Madiman-Barron inequality.

math.PR↗
arXiv2026

The maximal correlation coefficient associated with the minimum

For independent random variables $(X_i)_{1\leq i\leq n}$, we consider the maximal correlation coefficient $R=R(\min_{i:1\leq i\leq m}X_i,\min_{j:\ell+1\leq j\leq n}X_j)$. If $X_1,X_2,\ldots,X_n$ are identically distributed with the same continuous distribution, we find that $R=(m-\ell)/\sqrt{m(n-\ell)}$. For discrete distributions, we calculate the maximal correlation coefficient $R$ for Bernoulli distributions, geometric distributions, binomial distributions and Poisson distributions. Our paper answers a question in \cite[Section~6]{ChangChen}.

math.PR↗
arXiv2026

Littlewood-Offord problems for Ising models

We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration function \[Q_n(x,v)=P\left(\sum_{i=1}^{n}\varepsilon_iv_i\in(x-1,x+1)\right),\] where $x\in\mathbb{R}$, $v_1,v_2,\ldots,v_n$ are real numbers such that $|v_1|\geq 1, |v_2|\geq 1,\ldots, |v_n|\geq 1$, and $(\varepsilon_i)_{i=1,2,\ldots,n}\in\{-1,1\}^{n}$ are random spins of some Ising model. Let $Q_n=\sup_{x,v}Q_n(x,v)$. Under natural assumptions, we show that there exists a universal constant $C$ such that for all $n\geq 1$, \[\binom{n}{[n/2]}2^{-n}\leq Q_n\leq Cn^{-\frac{1}{2}}.\] As an application of the method, under the same assumption, we give a lower bound on the smallest eigenvalue of the truncated correlation matrix of the Ising model.

math.PR↗
arXiv2025

Strong law of large numbers for a function of the local times of a transient random walk on groups

This paper presents the strong law of large numbers for a function of the local times of a transient random walk on groups, extending the research of Asymont and Korshunov for random walks on the integer lattice $\mathbb{Z}^d$. Under some weaker conditions, we prove that certain function of the local times converges almost surely and in $L^1$ and $L^2$. The proof is mainly based on the subadditive ergodic theorem.

math.PR↗
arXiv2025

Reinforced Loop Soup via Wilson's Algorithm

The goal of this note is twofold: first, we explain the relation between the isomorphism theorems in the context of vertex reinforced jump process discovered in [BHS19, BHS21] and the standard Markovian isomorphism theorems for Markovian jump processes; second, we introduce the vertex reinforced counterpart of the standard Poissonian loop soup developed by Le Jan [LJ10]. To this end, we propose an algorithm that can be viewed as a variant of Wilson's algorithm with reinforcement. We establish the isomorphism theorems for the erased loops and the random walk from this algorithm, and in particular provide a concrete construction of the reinforced loop soup via a random process with a reinforcement mechanism.

math.PR↗
arXiv2025

A convergence result on the lengths of Markovian loops

Consider a sequence of Poisson point processes of non-trivial loops with certain intensity measures $(μ^{(n)})_n$, where each $μ^{(n)}$ is explicitly determined by transition probabilities $p^{(n)}$ of a random walk on a finite state space $V^{(n)}$ together with an additional killing parameter $c^{(n)}=e^{-a\cdot\sharp V^{(n)}+o(\sharp V^{(n)})}$. We are interested in asymptotic behavior of typical loops. Under general assumptions, we study the asymptotics of the length of a loop sampled from the normalized intensity measure $\barμ^{(n)}$ as $n\rightarrow\infty$. A typical loop is small for $a=0$ and extremely large for $a=\infty$. For $a=(0,\infty)$, we observe both small and extremely large loops. We obtain explicit formulas for the asymptotics of the mass of intensity measures, the asymptotics of the proportion of big loops, limit results on the number of vertices (with multiplicity) visited by a loop sampled from $\barμ^{(n)}$. We verify our general assumptions for random walk loop soups on discrete tori and truncated regular trees. Finally, we consider random walk loop soups on complete graphs. Here, our general assumptions are violated. In this case, we observe different asymptotic behavior of the length of a typical loop.

math.PR↗
arXiv2024

Littlewood-Offord problems for the Curie-Weiss models

In this paper, we consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. Let \[Q_n^{+}:=\sup_{x\in\mathbb{R}}\sup_{v_1,v_2,\ldots,v_n\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1)),\] \[Q_n=\sup_{x\in\mathbb{R}}\sup_{|v_1|,|v_2|,\ldots,|v_n|\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1))\] where the random variables $(\varepsilon_i)_{1\leq i\leq n}$ are spins in Curie-Weiss models. We calculate the asymptotic properties of $Q_n^{+}$ and $Q_n$ as $n\to\infty$ and observe the phenomena of phase transitions. Meanwhile, we also get that $Q_n^{+}$ is attained when $v_1=v_2=\cdots=v_n=1$. And $Q_n$ is attained when one half of $(v_i)_{1\leq i\leq n}$ equals to $1$ and the other half equals to $-1$ when $n$ is even.This is a generalization of classical Littlewood-Offord problems from Rademacher random variables to possibly dependent random variables. In particular, it includes the case of general independent and identically distributed Bernoulli random variables.

math.PR↗
arXiv2024

Greedy lattice paths with general weights

Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(π)=\sum_{v\inπ}X_v$ be the weight of a self-avoiding lattice path $π$. Let \[M_n=\max\{S(π):π\text{ has length }n\text{ and starts from the origin}\}.\] We are interested in the asymptotics of $M_n$ as $n\to\infty$. This model is closely related to the first passage percolation when the weights $\{X_v:v\in\mathbb{Z}^d\}$ are non-positive and it is closely related to the last passage percolation when the weights $\{X_v,v\in\mathbb{Z}^d\}$ are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that $\existsα>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+α}<+\infty$ and that $E[X_0^{-}]<+\infty$, we prove that there exists a finite real number $M$ such that $M_n/n$ converges to a deterministic constant $M$ in $L^{1}$ as $n$ tends to infinity. And under the stronger assumptions that $\existsα>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+α}<+\infty$ and that $E[(X_0^{-})^4]<+\infty$, we prove that $M_n/n$ converges to the same constant $M$ almost surely as $n$ tends to infinity.

math.PR↗
arXiv2024

Percolation threshold for metric graph loop soup

In this short note, we show that the critical threshold for the percolation of metric graph loop soup on a large class of transient metric graphs (including quasi-transitive graphs such as $\mathbb{Z}^d$, $d\geq 3$) is $1/2$.

math.PR↗
arXiv2022

Bernoulli hyper-edge percolation on Zd

We consider Bernoulli hyper-edge percolation on $\mathbb{Z}^d$. This model is a generalization of Bernoulli bond percolation. An edge connects exactly two vertices and a hyper-edge connects more than two vertices. As in the classical Bernoulli bond percolation, we open hyper-edges independently in a homogeneous manner with certain probabilities parameterized by a parameter $u\in[0,1]$. We discuss conditions for non-trivial phase transitions when $u$ varies. We discuss the conditions for the uniqueness of the infinite cluster. Also, we provide conditions under which the Grimmett-Marstrand type theorem holds in the supercritical regime.

math.PR↗
arXiv2020

On $H^{2|2}$ Isomorphism theorems and reinforced loop soup

We show that supersymmetric (susy) hyperbolic isomorphism theorems that relate Vertex Reinforced Jump Processes and $H^{2|2}$ field, introduced in [2] and [3], are annealed version of isomorphism theorems relating Markov processes and Gaussian free field, with the help of a Bayes formula that relates susy hyperbolic field to susy free field. On the other hand, we also prove a BFS-Dynkin's isomorphism theorem for reinforced loop soup. Moreover, we provide yet another proof of BFS-Dynkin's isomorphism for VRJP a la Feynman-Kac.

math.PR↗
arXiv2019

Second order asymptotics for Krein indefinite multipliers with multiplicity two

We consider linear Hamiltonian equations in $\mathbb{R}^{4}$ of the following type \begin{equation} \frac{\mathrm{d}γ}{\mathrm{d}t}(t)=J_{4}A(t)γ(t), γ(0)\in\operatorname{Sp}(4,\mathbb{R}), \end{equation} where $J=J_{4}\overset{\text{def}}{=}\begin{bmatrix}0 & \operatorname{Id}_2\\-\operatorname{Id}_2 & 0\end{bmatrix}$ and $A:t\mapsto A(t)$ is a $C^1$-continuous curve in the space of $4\times 4$ real matrices which are symmetric. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with multiplicity two.

math.DS↗
arXiv2017

On bifurcation of eigenvalues along convex symplectic paths

We consider a continuously differentiable curve $t\mapsto γ(t)$ in the space of $2n\times 2n$ real symplectic matrices, which is the solution of the following ODE: $\frac{\mathrm{d}γ}{\mathrm{d}t}(t)=J_{2n}A(t)γ(t), γ(0)\in\operatorname{Sp}(2n,\mathbb{R})$, where $J=J_{2n}\overset{\text{def}}{=}\begin{bmatrix}0 & \operatorname{Id}_n\\-\operatorname{Id}_n & 0\end{bmatrix}$ and $A:t\mapsto A(t)$ is a continuous in the space of $2n\times2n$ real matrices which are symmetric. Under certain convexity assumption (which includes the particular case that $A(t)$ is strictly positive definite for all $t\in\mathbb{R}$), we investigate the dynamics of the eigenvalues of $γ(t)$ when $t$ varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein-Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland: $\{t\in\mathbb{R}:γ(t)\text{ has a Krein indefinite eigenvalue of modulus }1\}$ is a discrete set.

math.DS↗
arXiv2015

Supercritical loop percolation on $\mathbb{Z}^d$ for $d\geq 3$

In this paper, we are interested in the loop cluster model on $\mathbb{Z}^d$ for $d\geq 3$. It is a long range model with two parameters $α$ and $κ$, where the non-negative parameter $α$ measures the amount of loops, and $κ$ plays the role of killing on vertices penalizing ($κ\geq 0$) or favoring ($κ<0$) appearance of large loops. We consider the truncated loop cluster model formed by the Poisson point process $\mathcal{L}_{α,\leq m}$, which is the restriction of $\mathcal{L}_α$ on loops with at most $m$ jumps. We prove the existence of percolation in a $2$-dimensional slab for the truncated loop model $\mathcal{L}_{α,\leq m}$ as long as the intensity parameter $α$ is strictly above the critical threshold of the non-truncated loop model and $m$ is large enough. We apply this result to prove the exponential decay of one arm connectivity for the finite cluster at $0$ for the whole supercritical regime of the non-truncated loop model. For $κ=0$, this loop percolation model provides an example in which we have different behaviors of finite clusters in sub-critical and super-critical regimes. Also, we deduce the strict increase of the critical curve $α\rightarrowκ_c(α)$ for $α\geqα_c$, where $α_c$ is the critical value when $κ=0$. In the end, we prove that $\forallα>α_c$ large balls in the infinite cluster are finally very regular in the sense of \cite{Sapozhnikov2014}, which implies that large balls are finally very good in the sense of \cite{BarlowMR2094438}. By \cite{BarlowMR2094438} and \cite{BarlowHamblyMR2471657}, we have Harnack's inequality and Gaussian type estimate for simple random walks on the infinite cluster for all $α>α_c$.

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