arXiv subjects

Zherui Fan

Publications and source records attributed to Zherui Fan.

6 recordsLinked to original sources

arXiv2025

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

By recent works of B\"aumler [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

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 $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $\beta>0$ 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 $2/(1+\delta)$, where $\delta\in (0,1)$ 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↗

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 $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $\beta>0$ 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 $n^{\delta(\beta)}$ for some $\delta(\beta)\in (0,1)$.

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 $1-\exp\{-\beta |i-j|^{-2}\}$ for some fixed $\beta>0$. 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 $\beta>0$ and thus rules out a potential phase transition (around $\beta = 1$) which seemed to be a reasonable possibility.

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 B\"aumler [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

Roughness of geodesics in Liouville quantum gravity

The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In the current article we confirm the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is stirctly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter $\xi$. We deduce this from a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). We also prove an analogous result for the LFPP geodesics.

math.PR↗