arXiv subjects

Maarten Markering

Publications and source records attributed to Maarten Markering.

7 recordsLinked to original sources

arXiv2026

The scaling limit of random walk and the intrinsic metric on planar critical percolation

We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $\kappa = 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations.

math.PR↗

arXiv2024

Capacity of loop-erased random walk

We study the capacity of loop-erased random walk (LERW) on $\mathbb{Z}^d$. For $d\geq4$, we prove a strong law of large numbers and give explicit expressions for the limit in terms of the non-intersection probabilities of a simple random walk and a two-sided LERW. Along the way, we show that four-dimensional LERW is ergodic. For $d=3$, we show that the scaling limit of the capacity of LERW is random. We show that the capacity of the first $n$ steps of LERW is of order $n^{1/\beta}$, with $\beta$ the growth exponent of three-dimensional LERW. We express the scaling limit of the capacity of LERW in terms of the capacity of Kozma's scaling limit of LERW. As a corollary, we obtain the scaling limit of the LERW in three dimensions when parametrized by its capacity.

math.PR↗

arXiv2024

Two-sided infinite self-avoiding walk in high dimensions

We construct the two-sided infinite self-avoiding walk (SAW) on $\mathbb{Z}^d$ for $d\geq5$ and use it to prove pattern theorems for the self-avoiding walk. We show that infinite two-sided SAW is the infinite-shift limit of infinite one-sided SAW and the infinite-size limit of finite two-sided SAW. We then prove that for every pattern $\zeta$, the fraction of times $\zeta$ occurs in the SAW converges to the probability that the two-sided infinite SAW starts with $\zeta$. The convergence is in probability for the finite SAW and almost surely for the infinite SAW. Along the way, we show that infinite SAW is ergodic using a coupling technique. At the end of the paper, we pose a conjecture regarding the existence of infinite SAW in low dimensions. We show that this conjecture is true in high dimensions, thus giving a new proof for the existence of infinite SAW for $d\geq5$. The proofs in this paper rely only on the asymptotics for the number of self-avoiding paths and the SAW two-point function. Although these results were shown by Hara and Slade using the lace expansion, the proofs in this paper do not use the lace expansion and might be adapted to prove existence and ergodicity of other infinite high-dimensional lattice models.

math.PR↗

arXiv2023

Cover times for random walk on dynamical percolation

We study the cover time of random walk on dynamical percolation on the torus $\mathbb{Z}_n^d$ in the subcritical regime. In this model, introduced by Peres, Stauffer and Steif, each edge updates at rate $\mu$ to open with probability $p$ and closed with probability $1-p$. The random walk jumps along each open edge with rate $1/(2d)$. We prove matching (up to constants) lower and upper bounds for the cover time, which is the first time that the random walk has visited all vertices at least once. Along the way, we also obtain a lower bound on the hitting time of an arbitrary vertex starting from stationarity, improving on the maximum hitting time bounds by Peres, Stauffer and Steif.

math.PR↗

arXiv2023

Large deviation principle for the norm of the Laplacian matrix of inhomogeneous Erd\H{o}s-R\'enyi random graphs

We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r_N(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r_N\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $\lambda_N$ be the maximal eigenvalue of the Laplacian matrix of $G_N$. We show that if $\lim_{N\to\infty} \|r_N-r\|_\infty = 0$ for some limiting graphon $r\colon\,[0,1]^2 \to (0,1)$, then $\lambda_N/N$ satisfies a downward LDP with rate $\binom{N}{2}$ and an upward LDP with rate $N$. We identify the associated rate functions $\psi_r$ and $\widehat{\psi}_r$, and derive their basic properties.

math.PR↗

arXiv2021

Breaking of ensemble equivalence for dense random graphs under a single constraint

Two ensembles are often used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. The latter means that it matters for the scaling properties of the graph whether the constraint is met for every single realisation of the graph or only holds as an ensemble average. In the literature, it was found that BEE is the rule rather than the exception for two classes: sparse random graphs when the number of constraints is of the order of the number of vertices and dense random graphs when there are two or more constraints that are frustrated. In the present paper we establish BEE for a third class: dense random graphs with a single constraint, namely, on the density of a given finite simple graph. We show that BEE occurs only in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as spectral signature of BEE. Proofs are based on an analysis of the variational formula on the space of graphons for the limiting specific relative entropy derived by Den Hollander et al. (2018), in combination with an identification of the minimising graphons and replica symmetry arguments. We show that in the replica symmetric region of the BEE-phase, as the size of the graph tends to infinity, the microcanonical ensemble behaves like an Erd\H{o}s-R\'enyi random graph, while the canonical ensemble behaves like a mixture of two Erd\H{o}s-R\'enyi random graphs.

math.PR↗

arXiv2020

The large deviation principle for inhomogeneous Erd\H{o}s-R\'enyi random graphs

Consider the inhomogeneous Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices for which each pair $i,j\in\{1,\ldots,n\}$, $i\neq j$ is connected independently by an edge with probability $r_n(\frac{i-1}{n},\frac{j-1}{n})$, where $(r_n)_{n\in\mathbb{N}}$ is a sequence of graphons converging to a reference graphon $r$. As a generalization of the celebrated LDP for ERRGs by Chatterjee and Varadhan (2010), Dhara and Sen (2019) proved a large deviation principle (LDP) for a sequence of such graphs under the assumption that $r$ is bounded away from 0 and 1, and with a rate function in the form of a lower semi-continuous envelope. We further extend the results by Dhara and Sen. We relax the conditions on the reference graphon to $\log r,\log(1-r)\in L^1([0,1]^2)$. We also show that, under this condition, their rate function equals a different, more tractable rate function. We then apply these results to the large deviation principle for the largest eigenvalue of inhomogeneous ERRGs and weaken the conditions for part of the analysis of the rate function by Chakrabarty, Hazra, Den Hollander and Sfragara (2020).

math.PR↗