arXiv subjects

Sara Terveer

Publications and source records attributed to Sara Terveer.

6 recordsLinked to original sources

arXiv2026

Return probability on Bienaym\'e-Galton-Watson trees and spectral asymptotics of sparse Erd\H{o}s-R\'enyi random graphs

We derive an upper bound for the annealed return probability of the simple random walk on supercritical Bienaym\'e-Galton-Watson trees. The bound decays subexponentially in time $t$ with $t^{1/3}$ in the exponent. It is valid for all offspring distributions with a finite first moment and is optimal whenever the offspring distribution does not exclude leaves or linear pieces in the tree. This solves completely the cases left open by Piau [Ann. Probab. 26, 1016-1040 (1998)]. A new feature of our proof is a far-reaching flexibility in the location of regions with bad isoperimetric properties in the tree. It allows to efficiently treat general offspring distributions and is gained from the joint consideration of the random tree and the random walk as it is inherent under the annealed measure. In the special case of a Poissonian offspring distribution we apply the upper bound for the annealed return probability to deduce a Lifshits tail for the empirical eigenvalue distribution of the graph Laplacian on supercritical Erd\H{o}s--R\'enyi random graphs with finite mean degree.

math.PR↗

arXiv2025

Spectral properties of the stochastic block model and their application to hitting times of random walks

We analyze hitting times of simple random walk on realizations of the stochastic block model. We show that under some natural assumptions the hitting time averaged over the target vertex asymptotically almost surely given by $N(1+o(1))$. On the other hand, the hitting time averaged over the starting vertex asymptotically almost surely depends on expected degrees in the block the target vertex is in. We also show a central limit theorem for the hitting time averaged over the starting vertex. Our main techniques are a spectral decomposition of these hitting times, a spectral analysis of the adjacency matrix and the graph Laplacian.

math.PR↗

arXiv2025

Random walks with square-root boundaries: the case of exact boundaries $g(t)=c\sqrt{t+b}-a$

Let $S(n)$ be a real valued random walk with i.i.d. increments which have zero mean and finite variance. We are interested in the asymptotic properties of the stopping time $T(g):=\inf\{n\ge1: S(n)\le g(n)\}$, where $g(t)$ is a boundary function. In the present paper we deal with the parametric family of boundaries $\{g_{a,b}(t)=c\sqrt{t+b}-a, b\ge0, a>c\sqrt{b}\}$. First, assuming that sufficiently many moments of increments of the walk are finite, we construct a positive space-time harmonic function $W(a,b)$. Then we show that there exist $p(c)>0$ and a constant $\varkappa(c)$ such that $\mathbf{P}(T_{g_{a,b}}>n)\sim \varkappa(c)\frac{W(a,b)}{n^{p(c)/2}}$ as $n\to\infty$.

math.PR↗

arXiv2023

A Central Limit Theorem for the average target hitting time for a random walk on a random graph

Consider a simple random walk on a realization of an Erdős-Rényi graph. Assume that it is asymptotically almost surely (a.a.s.) connected. Conditional on an eigenvector delocalization conjecture, we prove a Central Limit Theorem (CLT) for the average target hitting time. By the latter we mean the expected time it takes the random walk on average to first hit a vertex $j$ when starting in a fixed vertex $i$. The average is taken with respect to $π_i$, the invariant measure of the random walk.

math.PR↗

arXiv2020

A Central Limit Theorem for the mean starting hitting time for a random walk on a random graph

We consider simple random walk on a realization of an Erdős-Rényi graph that is asymptotically almost surely (a.a.s.) connected. We show a Central Limit Theorem (CLT) for the average starting hitting time, i.e. the expected time it takes the random walker on average to first hit a vertex $j$ when starting in a fixed vertex $i$. The average is taken with respect to $π_j$, the invariant measure of the random walk.

math.PR↗

arXiv2020

A Central Limit Theorem for incomplete U-statistics over triangular arrays

We analyze the fluctuations of incomplete $U$-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled and centered version of the U-statistic converges to a normal random variable. Our method of proof relies on a martingale CLT. A possible application -- a CLT for the hitting time for random walk on random graphs -- will be presented in \cite{LoTe20b}

math.PR↗