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Lu-Jing Huang

Publications and source records attributed to Lu-Jing Huang.

15 recordsLinked to original sources
arXiv2025

Uniqueness and dimension for the geodesic of the critical long-range percolation metric

By recent works of Bäumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points, for which an important ingredient of independent interest is the continuity of the metric distribution. In addition, we establish the Hausdorff dimension of the geodesics.

math.PR↗
arXiv2025

The polynomial growth of effective resistances in one-dimensional critical long-range percolation

We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both grow like $n^{δ(β)}$ for some $δ(β)\in (0,1)$.

math.PR↗
arXiv2025

Spectral dimensions for one-dimensional critical long-range percolation

Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+δ)$, where $δ\in (0,1)$ is the exponent of the effective resistance in the LRP model, as derived in [10, Theorem 1.1]. Our work addresses an open question from [7, Section 5].

math.PR↗
arXiv2024

Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation

In this work, we study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β|i-j|^{-2}\}$ for some fixed $β>0$. Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to $[-N, N]^c$ and from the interval $[-N,N]$ to $[-2N,2N]^c$ (conditioned on no edge joining $[-N,N]$ and $[-2N,2N]^c$) both have a polynomial lower bound in $N$. Our bound holds for all $β>0$ and thus rules out a potential phase transition (around $β= 1$) which seemed to be a reasonable possibility.

math.PR↗
arXiv2024

Symmetry and functional inequalities for stable Lévy-type operators

In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$\mathcal{L}=a(x){Δ^{α/2}}+b(x)\frac{\d}{\d x},$$ where $a,b$ are the continuous positive and differentiable functions, respectively. Under the assumption of symmetry, we further study the criteria for functional inequalities, including Poincaré inequalities, logarithmic Sobolev inequalities and Nash inequalities. Our proofs rely on the Orlicz space theory and the estimates of the Green functions.

math.PR↗
arXiv2023

Uniqueness of the critical long-range percolation metrics

In this work, we study the random metric for the critical long-range percolation on $\mathbb{Z}^d$. A recent work by Bäumler [3] implies the subsequential scaling limit, and our main contribution is to prove that the subsequential limit is uniquely characterized by a natural list of axioms. Our proof method is hugely inspired by recent works of Gwynne and Miller [42], and Ding and Gwynne [25] on the uniqueness of Liouville quantum gravity metrics.

math.PR↗
arXiv2022

Dirichlet eigenvalues and exit time moments for symmetric Markov processes

We give some relationships between the first Dirichlet eigenvalues and the exit time moments for the general symmetric Markov processes. As applications, we present some examples, including symmetric diffusions and $α$-stable processes, and provide the estimates of their first Dirichlet eigenvalues and the exit time moments.

math.PR↗
arXiv2021

Variational formulas for the exit time of Hunt processes generated by semi-Dirichlet forms

Variational formulas for the Laplace transform of the exit time from an open set of a Hunt process generated by a regular lower bounded semi-Dirichlet form are established. While for symmetric Markov processes, variational formulas are derived for the exponential moments of the exit time. As applications, we provide some comparison theorems and quantitative relations of the exponential moments and Poincaré inequalities.

math.PR↗
arXiv2021

Strict Kantorovich contractions for Markov chains and Euler schemes with general noise

We study contractions of Markov chains on general metric spaces with respect to some carefully designed distance-like functions, which are comparable to the total variation and the standard $L^p$-Wasserstein distances for $p \ge 1$. We present explicit lower bounds of the corresponding contraction rates. By employing the refined basic coupling and the coupling by reflection, the results are applied to Markov chains whose transitions include additive stochastic noises that are not necessarily isotropic. This can be useful in the study of Euler schemes for SDEs driven by Lévy noises. In particular, motivated by recent works on the use of heavy tailed processes in Markov Chain Monte Carlo, we show that chains driven by the $α$-stable noise can have better contraction rates than corresponding chains driven by the Gaussian noise, due to the heavy tails of the $α$-stable distribution.

math.PR↗
arXiv2020

Variational Formulas of Asymptotic Variance for General Discrete-time Markov Chains

The asymptotic variance is an important criterion to evaluate the performance of Markov chains, especially for the central limit theorems. We give the variational formulas for the asymptotic variance of discrete-time (non-reversible) Markov chains on general state space. The variational formulas provide many applications, extending the classical Peskun's comparison theorem to non-reversible Markov chains, and obtaining several comparison theorems between Markov chains with various perturbations.

math.PR↗
arXiv2020

Approximation of heavy-tailed distributions via stable-driven SDEs

Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this paper we provide a rigorous theoretical framework for studying the problem of approximating heavy-tailed distributions via ergodic SDEs driven by symmetric (rotationally invariant) $α$-stable processes.

math.PR↗
arXiv2020

Capacity and Exit Time for Non-reversible Diffusions

Capacity is an important quantity in potential theory and in the study of Markov processes. We give equivalent conditions between the capacity, the mean exit time, and the Green function for non-reversible diffusions.

math.PR↗
arXiv2019

On hitting time, mixing time and geometric interpretations of Metropolis-Hastings reversiblizations

Given a target distribution $μ$ and a proposal chain with generator $Q$ on a finite state space, in this paper we study two types of Metropolis-Hastings (MH) generator $M_1(Q,μ)$ and $M_2(Q,μ)$ in a continuous-time setting. While $M_1$ is the classical MH generator, we define a new generator $M_2$ that captures the opposite movement of $M_1$ and provide a comprehensive suite of comparison results ranging from hitting time and mixing time to asymptotic variance, large deviations and capacity, which demonstrate that $M_2$ enjoys superior mixing properties than $M_1$. To see that $M_1$ and $M_2$ are natural transformations, we offer an interesting geometric interpretation of $M_1$, $M_2$ and their convex combinations as $\ell^1$ minimizers between $Q$ and the set of $μ$-reversible generators, extending the results by Billera and Diaconis (2001). We provide two examples as illustrations. In the first one we give explicit spectral analysis of $M_1$ and $M_2$ for Metropolised independent sampling, while in the second example we prove a Laplace transform order of the fastest strong stationary time between birth-death $M_1$ and $M_2$.

math.PR↗