arXiv subjects

Chenlin Gu

Publications and source records attributed to Chenlin Gu.

At least 19 recordsLinked to original sources

arXiv2026

Branching random walk in random environment

We consider a branching random walk on $\Z^d$ in a random environment given by Bernoulli site percolation with parameter $p\in (0,1)$. In this model, each particle located at an open site reproduces according to a law $\mu_\circ$, whereas a particle at a closed site reproduces according to another law $\mu_\bullet$. Each newly born child performs an independent simple random walk jump from the position of its parent. We study the quenched survival probability under various assumptions on $(\mu_\circ, \mu_\bullet)$ and establish a Yaglom theorem when both offspring distributions are critical.

math.PR↗

arXiv2026

Quadratic fluctuations of speed-change Kawasaki dynamics

For the speed-change Kawasaki dynamics, we study the weak convergence of its quadratic field, and derive the equilibrium fluctuation. This extends the result of Gon{\c{c}}alves and Jara [ALEA, Lat. Am. J. Probab. Math. Stat. 16, 605-632 (2019)] to the non-gradient case.

math.PR↗

arXiv2026

Perron--Frobenius theorem for a general tree-valued growth-fragmentation-isolation process

A general tree-valued dynamics is considered in continuous time: new vertices are added, and the percolation happens on the links, and the connected components can be frozen. The model is an infinite-type branching process. The main result establishes the Perron--Frobenius type theorem on this model, which extends the previous work [Ann. Appl. Probab. 33 (6B) 5233 - 5278]. The proof does not rely on any property of the uniform random recursive tree.

math.PR↗

arXiv2026

Scaling limit of additive functionals for reversible non-gradient exclusion process: critical cases

For the reversible speed-change exclusion process $(\eta_t)_{t \geq 0}$ in $\mathbb{Z}^d$, we study the scaling limit of additive functionals ${\Gamma_t(f) = \int_0^t f(\eta_s)\, \mathrm{d} s}$. Concerning the local centered function $f$, the previous work [Commun. Math. Phys. 104, 1-19, 1986] by Kipnis and Varadhan and [Comm. Pure Appl. Math., 66: 649-677, 2013] by Gon{\c{c}}alves and Jara respectively covered the cases $d \geq 3$ and $d=1$. The present paper completes the missing part $d=2$, and also develops the theory for functions with higher degree. The novelty is a quantitative homogenization of the resolvent, which allows to overcome the obstacle of correlation function in non-gradient models.

math.PR↗

arXiv2026

Quantitative stochastic homogenization for long-range random walks with critical jump index

In this paper, we study the stochastic homogenization for a class of symmetric random walks in random conductance model, whose one-step transition probability from $x$ to $y$ is proportional to $|x-y|^{-d-2}$. As the associated jumping kernel fails to be $L^2$-integrable yet admits a finite $\alpha$-th moment for all $\alpha\in (0,2)$, we refer to the corresponding process $(X^\w_t)_{t\ge0}$ as a long-range random walk with critical jump index. In this critical regime, the scaled process $\bigl(k^{-1}X_{k^2(\log k)^{-1}t}\bigr)_{t\ge 0}$, whose scaling order is different from the diffusive scaling and the $\alpha$-stable scaling, converges to a Brownian motion. Besides characterizing the limiting Brownian motion, we will give a convergence rate for associated scaled resolvents, which obeys the order $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$.

math.PR↗

arXiv2025

Relaxation to equilibrium of conservative dynamics II: non-gradient exclusion processes

For the speed-change exclusion process on $\mathbb{Z}^d$ reversible with respect to the product Bernoulli measure, we prove that its semigroup $P_t$ satisfies a variance decay $\operatorname{Var}[P_t u] = C_u t^{-\frac{d}{2}} + o(t^{-\frac{d+\delta}{2}})$ for every local function $u$, with the constant $C_u$ explicitly characterized. This extends the result of Janvresse, Landim, Quastel and Yau in [Ann. Probab. 27(1) 325--360, 1999] to a non-gradient model. The proof combines the regularization argument in the previous work, and the chaos expansion in [Markov Process. Related Fields, 5(2) 125--162, 1999] by Bertini and Zegarlinski, via a new input from the homogenization theory.

math.PR↗

arXiv2025

The diffusivity of supercritical Bernoulli percolation is infinitely differentiable

We prove that, the diffusivity and conductivity on $\mathbb{Z}^d$-Bernoulli percolation ($d \geq 2$) are infinitely differentiable in supercritical regime. This extends a result by Kozlov [Uspekhi Mat. Nauk 44 (1989), no. 2(266), pp 79 - 120]. The key to the proof is a uniform estimate for the finite-volume approximation of derivatives, which relies on the perturbed corrector equations in homogenization theory. The renormalization of geometry is then implemented in a sequence of scales to gain sufficient degrees of regularity. To handle the higher-order perturbation on percolation, new techniques, including cluster-growth decomposition and hole separation, are developed.

math.PR↗

arXiv2024

Coupling between Brownian motion and random walks on the infinite percolation cluster

For the supercritical Bernoulli bond percolation on $\mathbb{Z}^d$ ($d \geq 2$), we give a coupling between the random walk on the infinite cluster and its limit Brownian motion, such that the maximum distance between the paths during $[0,T]$ has a mean of order $T^{\frac{1}{3}+o(1)}$. The construction of the coupling utilizes the optimal transport tool. The analysis mainly relies on local CLT and the concentration of the cluster density. This partially answers an open question posed by Biskup [Probab. Surv., 8:294-373, 2011]. As a direct application, our result recovers the law of the iterated logarithm proved by Duminil-Copin [arXiv:0809.4380], and further identifies the limit constant.

math.PR↗

arXiv2024

Size distribution of clusters in site-percolation on random recursive tree

We prove rigorously several results about the site-percolation on random recursive trees, observed in the previous work by Kalay and Ben-Naim [J. Phys. A48(2015), no.4, 0405001, 15 pp.]. For a random recursive tree of size $n$, let every site have probability ${p \in (0,1)}$ to remain and with probability $(1-p)$ to be removed. As $n\to\infty,$ we show that the proportion of the remaining clusters of size $k$ is of order $k^{-1-\frac{1}{p}}$, resulting in a Yule-Simon distribution; the largest cluster size is of order $n^{p}$, and admits a non-trivial scaling limit. The proofs are based on the embedding of this model in the multi-type branching processes, and a coupling with the bond-percolation on random recursive trees.

math.PR↗

arXiv2024

Speed of random walk on dynamical percolation in nonamenable transitive graphs

Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $\mu>0$, and following the refresh, each edge is open independently with probability $p$. The random walk traverses $G$ only along open edges, moving at rate $1$. In the critical regime $p=p_c$, we prove that the speed of the random walk is at most $O(\sqrt{\mu \log(1/\mu)})$, provided that $\mu \le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $\mu)$, while in the subcritical regime $p<p_c$, the speed is of order $\mu\wedge 1$.

math.PR↗

arXiv2024

Random walk on dynamical percolation in Euclidean lattices: separating critical and supercritical regimes

We study the random walk on dynamical percolation of $\mathbb{Z}^d$ (resp., the two-dimensional triangular lattice $\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $\mu\in (0,1/e]$. The random walk moves along open edges in $\mathbb{Z}^d$ (resp., open sites in $\mathcal{T}$) at rate $1$. For the critical regime $p=p_c$, we prove the following two results: on $\mathcal{T}$, the mean squared displacement of the random walk from $0$ to $t$ is at most $O(t\mu^{5/132-\epsilon})$ for any $\epsilon>0$; on $\mathbb{Z}^d$ with $d\geq 11$, the corresponding upper bound for the mean squared displacement is $O(t \mu^{1/2}\log(1/\mu))$. For the supercritical regime $p>p_c$, we prove that the mean squared displacement on $\mathbb{Z}^d$ is at least $ct$ for some $c=c(d)>0$ that does not depend on $\mu$.

math.PR↗

arXiv2024

Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process

For the non-gradient exclusion process, we prove the quantitative homogenization of the diffusion matrix and the conductivity by local functions. The proof relies on the renormalization approach developed by Armstrong, Kuusi, Mourrat, and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse-grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. We then strengthen the convergence rate to be uniform concerning the density, and integrate it into the work by Funaki, Uchiyama, and Yau [IMA Vol. Math. Appl., 77 (1996), pp. 1-40.] to yield a quantitative hydrodynamic limit. Our new approach avoids showing the characterization of closed forms and provides stronger results. The extension is discussed for the model in the presence of disorder on the bonds.

math.PR↗

arXiv2024

Quantitative equilibrium fluctuations for interacting particle systems

We consider a class of interacting particle systems in continuous space of non-gradient type, which are reversible with respect to Poisson point processes with constant density. For these models, a rate of convergence was recently obtained in 10.1214/22-AOP1573 for certain finite-volume approximations of the bulk diffusion matrix. Here, we show how to leverage this to obtain quantitative versions of a number of results capturing the large-scale fluctuations of these systems, such as the convergence of two-point correlation functions and the Green-Kubo formula.

math.PR↗

arXiv2021

Smoothness of the diffusion coefficients for particle systems in continuous space

For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of correctors.

math.PR↗

arXiv2021

Quantitative homogenization of interacting particle systems

For a class of interacting particle systems in continuous space, we show that finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. Our approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, we develop suitable modifications of the Caccioppoli and multiscale Poincaré inequalities, which are of independent interest.

math.PR↗

arXiv2021

Power law decay at criticality for the q-state antiferromagnetic Potts model on regular trees

We present a proof of the power law decay of magnetic moment for the $q$-state antiferromagnetic Potts model on the regular tree at the critical temperature, and also justify that the exact exponent is $\frac{1}{2}$. Our proof relies on the assumption of the uniqueness at the critical temperature, which has been established for $q=3,4$, and for $q \ge 5$ with large degree. An iterative contraction inequality is developed for independent interests.

math.PR↗

arXiv2021

A growth-fragmentation-isolation process on random recursive trees and contact tracing

We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate proportional to their sizes (isolation of connected component). A phase transition occurs when the isolation is able to stop the growth fragmentation process and cause extinction. When the process survives, the number of clusters increases exponentially and we prove that the normalized empirical measure of clusters a.s. converges to a limit law on recursive trees. We exploit the branching structure associated with the size of clusters, which is inherited from the splitting property of random recursive trees. This work is motivated by the control of epidemics and contact tracing where clusters correspond to trees of infected individuals that can be identified and isolated. We complement this work by providing results on the Malthusian exponent to describe the effect of control policies on epidemics.

math.PR↗

arXiv2021

Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients

We study the iterative algorithm proposed by S. Armstrong, A. Hannukainen, T. Kuusi, J.-C. Mourrat to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this iterative method is comparatively easy to compute. In this article, we strengthen the estimate for the contraction factor achieved by one iteration of the algorithm. We obtain an estimate that holds uniformly over the initial function in the iteration, and which grows only logarithmically with the size of the domain.

math.NA↗