arXiv subjects

Yushu Zheng

Publications and source records attributed to Yushu Zheng.

10 recordsLinked to original sources

arXiv2026

Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$

We obtain sharp asymptotic probabilities for upward deviations of the maximum local time of discrete- and continuous-time simple random walks on $\mathbb{Z}^d$, $d\ge 3$. For downward deviations, we prove the sharp continuous-time asymptotics and the discrete-time upper bound. Together with the loop-pruning paper~\cite{li2026loopprune_inprep}, which proves the matching discrete-time lower bound via a loop-pruning construction, this yields the sharp downward-deviation asymptotics in discrete time as well. We also derive Gumbel-type consequences at the logarithmic scale.

math.PR↗

arXiv2026

Gaussian rigidity for infinite exchangeable sequences

We prove a Gaussian rigidity theorem for infinite exchangeable sequences of real-valued random variables: the joint Gaussianity of a single pair of entries already forces the entire sequence to be a Gaussian process. This settles a conjecture raised by Newman (2026). The main analytic ingredient in the proof is Hardy's uncertainty principle. We also obtain a finite-dimensional vector-valued extension.

math.PR↗

arXiv2026

Can the root cluster remain largest forever in random recursive tree percolation?

We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a largest table forever. We further show that two associated quantities, the limit as $n\to\infty$ of the probability that the root cluster is a largest cluster at time $n$ and the probability that it remains a largest cluster for all times, are strictly increasing and continuous functions of $p$ on $(0,1]$, and both tend to zero as $p\downarrow0$.

math.PR↗

arXiv2026

Loop pruning and downward deviations for maximum local time of discrete-time simple random walks

We study downward deviations of the maximum local time of the discrete-time simple random walk on $\mathbb{Z}^d$, $d\ge 3$. In our previous paper \cite{li2026ldmaxlocal}, the corresponding upper bound was established, while the matching lower bound was left open. In the present paper, we prove this lower bound and hence obtain the sharp asymptotic formula for the downward-deviation probability. To provide a discrete-time analogue of the jump-chain/holding-time structure used in the continuous-time argument, we introduce a new random structure which we name as {\it loop-pruned random walk} and the associated loop-pruning decomposition, which is also of independent interest.

math.PR↗

arXiv2026

From Cannings model to Brownian motion conditioned on local time profile

We study the scaling limits of genealogical trees arising from Cannings models. Under suitable moment conditions, we show that the rescaled contour and height functions converge to a time change of Brownian motion conditioned on a given local time profile. This conditioned Brownian motion is a self-interacting diffusion constructed independently by Warren--Yor (1998) and Aldous (1998). A key ingredient in our proof is a sequential version of the coming-down-from-infinity property.

math.PR↗

arXiv2025

Favorite sites for simple random walk in two and more dimensions

On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987), which was brought up again in Dembo (2005).

math.PR↗

arXiv2023

A note on $α$-permanent and loop soup

In this paper, it is shown that $α$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $α$-permanents of the block matrices obtained from matrices associated to $*$-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.

math.PR↗

arXiv2023

Regular subspaces of symmetric stable processes

Roughly speaking, regular subspaces are regular Dirichlet forms that inherit the original forms with smaller domains. In this paper, regular subspaces of 1-dim symmetric $α$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $α\in [1,2]$. Moreover, for $α\in(1,2)$, the characterization of the regular subspaces is given. General 1-dim symmetric Lévy processes will also be investigated. It will be shown that whether it has proper regular subspaces is closely related to whether its sample paths have finite variation.

math.PR↗

arXiv2022

Inverting Ray-Knight identities on trees

In this paper, we first introduce the Ray-Knight identity and percolation Ray-Knight identity related to loop soup with intensity $α(\ge 0)$ on trees. Then we present the inversions of the above identities, which are expressed in terms of repelling jump processes. In particular, the inversion in the case of $α=0$ gives the conditional law of a Markov jump process given its local time field. We further show that the fine mesh limits of these repelling jump processes are the self-repelling diffusions \cite{Aidekon} involved in the inversion of the Ray-Knight identity on the corresponding metric graph. This is a generalization of results in \cite{2016Inverting,lupu2019inverting,LupuEJP657}, where the authors explore the case of $α=1/2$ on a general graph. Our construction is different from \cite{2016Inverting,lupu2019inverting} and based on the link between random networks and loop soups.

math.PR↗

arXiv2022

On the one-sided boundedness of the local discrepancy of $\{nα\}$-sequences

The main interest of this article is the one-sided boundedness of the local discrepancy of $α\in\mathbb{R}\setminus\mathbb{Q}$ on the interval $(0,c)\subset(0,1)$ defined by \[D_n(α,c)=\sum_{j=1}^n 1_{\{\{jα\}<c\}}-cn.\] We focus on the special case $c\in (0,1)\cap\mathbb{Q}$. Several necessary and sufficient conditions on $α$ for $(D_n(α,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set \[O_c=\{α\in \irr: (D_n(α,c)) \text{ is one-side bounded}\}.\]

math.NT↗