arXiv subjects

Jie-Xiang Zhu

Publications and source records attributed to Jie-Xiang Zhu.

12 recordsLinked to original sources

arXiv2026

The Maximum Number of Shortest Paths in Graphs

Benjamini and Tzalik obtained an upper bound on the number of shortest paths between two vertices at distance $t$ in a multigraph of maximum degree at most $Δ$, and proposed a conjecture on the sharp bound. In this paper, we develop a probabilistic counting argument based on probability distributions induced by random walks from the two endpoints. This approach yields a sharp bound for multigraphs and confirms their conjecture. We further determine the exact maximum for simple graphs and thus answer another question of Benjamini and Tzalik. We also investigate the equality cases, describing the structure of the subgraph formed by shortest paths between $x$ and $y$ and giving tight examples.

math.CO↗

arXiv2026

Counting degree-constrained orientations

We study the enumeration of graph orientations under local degree constraints. Given a finite graph $G = (V, E)$ and a family of admissible sets $\{\mathsf P_v \subseteq \mathbb{Z} : v \in V\}$, let $\mathcal N (G; \prod_{v \in V} \mathsf P_v)$ denote the number of orientations in which the out-degree of each vertex $v$ lies in $P_v$. We prove a general duality formula expressing $\mathcal N(G; \prod_{v \in V} \mathsf P_v)$ as a signed sum over edge subsets, involving products of coefficient sums associated with $\{\mathsf P_v\}_{v \in V}$, from a family of polynomials. Our approach employs gauge transformations, a technique rooted in statistical physics and holographic algorithms. We also present a probabilistic derivation of the same identity, interpreting the orientation-generating polynomial as the expectation of a random polynomial product. As applications, we obtain explicit formulas for the number of even orientations and for mixed Eulerian-even orientations on general graphs. Our formula generalizes a result of Borbényi and Csikvári on Eulerian orientations of graphs.

math.CO↗

arXiv2026

Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions

For any $1 < p < q < \infty$, we investigate fixed-time hypercontractive bounds from $L^p$ to $L^q$ of Poisson semigroups associated with the Ornstein--Uhlenbeck, Laguerre and Jacobi operators. We prove that, in the Ornstein--Uhlenbeck and Laguerre cases, the Poisson semigroups fail to be $L^p \to L^q$ bounded for any fixed $t > 0$. In contrast, for Jacobi operators with $α, β\ge -1/2$, the associated Poisson semigroups are ultracontractive, namely bounded from $L^1$ to $L^\infty$. More generally, we study Bernstein subordinations of these semigroups and show that fixed-time hypercontractivity is not stable under subordination. The analysis relies on quantitative $L^q$-estimates for the corresponding orthogonal polynomial eigenfunctions, together with a bilinear test with the exponential family.

math.CA↗

arXiv2026

The Gundy-Stein decomposition with explicit constants

Let $(\mathcal F_n)_{n\ge 1}$ be a filtration and let $f\ge0$ belong to $L^1(\mathcal F_\infty)$. For the martingale $f_n=\mathbb E[f\mid \mathcal F_n]$ and each $λ>0$ we prove a Gundy--Stein decomposition \[ f=g+h+k \] with explicit numerical constants. In the positive closed case the three parts satisfy explicit bounds, and the bounded part is bounded above by $λ$. We also prove a one-parameter form for the bounded part and two-point sharpness results, including a joint sharpness statement for arbitrary decompositions under the condition $0\le k\le λ$. We also obtain an exact four-term refinement of the decomposition, separating the bounded term into a stopped part and a conditional expectation term. As applications we obtain an explicit weak-type $(1,1)$ estimate for truncated martingale multipliers and a John--Nirenberg inequality for martingale $\mathrm{BMO}$ on atomic $α$-regular filtrations.

math.PR↗

arXiv2025

Convergence rates for the $p$-Wasserstein distance of the empirical measures of an ergodic Markov process

Let $X:=(X_t)_{t\geq 0}$ be an ergodic Markov process on $\real^d$, and $p>0$. We derive upper bounds of the $p$-Wasserstein distance between the invariant measure and the empirical measures of the Markov process $X$. For this we assume, e.g.\ that the transition semigroup of $X$ is exponentially contractive in terms of the $1$-Wasserstein distance, or that the iterated Poincaré inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.

math.PR↗

arXiv2024

Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{nα\}$

Fix an irrational number $α$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $φ$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random walk $\{S_n α\}_{n\ge 1}$ on the torus, where $\{\cdot\}$ denotes the fractional part. We study the long time asymptotic behaviour of the empirical measure of this random walk to the uniform distribution under the general $p$-Wasserstein distance. Our results show that the Wasserstein convergence rate depends on the Diophantine properties of $α$ and the Hölder continuity of the characteristic function $φ$ at the origin, and there is an interesting critical phenomenon that will occur. The proof is based on the PDE approach developed by L. Ambrosio, F. Stra and D. Trevisan in [2] and the continued fraction representation of the irrational number $α$.

math.PR↗

arXiv2024

Wasserstein asymptotics for empirical measures of diffusions on four dimensional closed manifolds

We identify the leading term in the asymptotics of the quadratic Wasserstein distance between the invariant measure and empirical measures for diffusion processes on closed weighted four-dimensional Riemannian manifolds. Unlike results in lower dimensions, our analysis shows that this term depends solely on the Riemannian volume of the manifold, remaining unaffected by the potential and vector field in the diffusion generator.

math.PR↗

arXiv2024

Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E} [{\mathbb W}_p^q(μ_T,μ)])^{\frac{1}{q}} (T \to \infty)$ uniformly in $(p,q)\in [1,\infty) \times (0,\infty)$, where $μ_T$ is the empirical measure of the diffusion process, $μ$ is the unique invariant probability measure, and ${\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\mathbb E} |T {\mathbb W}_2^2(μ_T,μ)-Ξ(T)|^q \to 0$ as $T\to\infty$ for any $q\ge 1$, where $Ξ(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.

math.PR↗

arXiv2023

Asymptotic behavior of Wasserstein distance for weighted empirical measures of diffusion processes on compact Riemannian manifolds

Let $(X_t)_{t \geq 0}$ be a diffusion process defined on a compact Riemannian manifold, and for $α> 0$, let $$ μ_t^{(α)} = \fracα{t^α} \int_{0}^{t} δ_{X_s} \, s^{α- 1} \mathrm{d} s $$ be the associated weighted empirical measure. We investigate asymptotic behavior of $\mathbb{E}^ν\big[ \mathrm{W}_2^2(μ_t^{(α)}, μ) \big]$ for sufficient large $t$, where $\mathrm{W}_2$ is the quadratic Wasserstein distance and $μ$ is the invariant measure of the process. In the particular case $α= 1$, our result sharpens the limit theorem achieved in [26]. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.

math.PR↗

arXiv2023

Weak type $(1, 1)$ of Riesz transform on some direct product manifolds with exponential volume growth

In this paper we are concerned with the Riesz transform on the direct product manifold ${\mathbb{H}}^n \times M$, where ${\mathbb{H}}^n$ is the $n$-dimensional real hyperbolic space and $M$ is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type $(1,1)$ property. In addition, we obtain the weak type $(1, 1)$ of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type $(1,1)$ boundedness of some operators considered in the work of Li, Sjögren and Wu [27].

math.CA↗

arXiv2021

Limit Theorems in Warsserstein Distance for Empirical Measures of Diffusion Processes on Riemannian Manifolds

Let $M$ be a compact connected Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $μ(d x):=e^{V(x)}d x$ is a probability measure, and let $\{λ_i\}_{i\ge 1} $ be all non-trivial eigenvalues of $-L$ with Neumann boundary condition if the boundary exists. Then the empirical measures $\{μ_t\}_{t>0}$ of the diffusion process generated by $L$ (with reflecting boundary if the boundary exists) satisfy $$ \lim_{t\to \infty} \big\{t \mathbb E^x [W_2(μ_{t},μ)^2]\big\}= \sum_{i=1}^\infty\frac 2 {λ_i^2}\ \text{ uniformly\ in\ } x\in M,$$ where $\mathbb E^x$ denotes the expectation for the diffusion process starting at point $x$, $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric. The limit is finite if and only if $d\le 3$, and in this case we derive the following central limit theorem: $$\lim_{t\to\infty} \sup_{x\in M} \Big|\mathbb P^x(t W_2(μ_{t},μ)^2<a)- \mathbb P\Big( \sum_{k=1}^\infty \frac{2ξ_k^2}{λ_k^2}<a\Big)\Big|=0, \ \ a\ge 0,$$ where $\mathbb P^x$ is the probability with respect to $\mathbb E^x$, and $\{ξ_k\}_{k\ge 1}$ are i.i.d. standard Gaussian random variables. Moreover, when $d\ge 4$ we prove that the main order of $\mathbb E^x[W_2(μ_{t},μ)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Moreover, when $d\ge 4$ the main order of $\mathbb E^x[W_2(μ_{t},μ)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Finally, we establish the long-time large deviation principle for $\{W_2(μ_t,μ)^2\}_{t\ge 0}$ with a good rate function given by the information with respect to $μ$.

math.PR↗

arXiv2019

On optimal matching of Gaussian samples III

This article is a continuation of the papers [8,9] in which the optimal matching problem, and the related rates of convergence of empirical measures for Gaussian samples are addressed. A further step in both the dimensional and Kantorovich parameters is achieved here, proving that, given $X_1, \ldots, X_n$ independent random variables with common distribution the standard Gaussian measure $μ$ on $\mathbb{R}^d$, $d \geq 3$, and $μ_n \, = \, \frac 1n \sum_{i=1}^n δ_{X_i}$ the associated empirical measure, $$ \mathbb{E} \big( \mathrm {W}_p^p (μ_n , μ)\big ) \, \approx \, \frac {1}{n^{p/d}} $$ for any $1\leq p < d$, where $\mathrm {W}_p$ is the $p$-th Kantorovich metric. The proof relies on the pde and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact setting.

math.PR↗