arXiv subjects

Yuval Peres

Publications and source records attributed to Yuval Peres.

At least 19 recordsLinked to original sources

arXiv2026

Mixing time under monotone censoring

We prove the conjecture of Ding and Mossel~\cite{DM}: the lazy random walk on the discrete cube, censored to any increasing set of fixed positive density, mixes in time $O(n\log n)$. More precisely, for every nonempty increasing set $A\subseteq\{0,1\}^n$, with $a=μ(A)$ and $μ$ uniform on the cube, we prove $$ t_{\mathrm{mix}}(P)\le C_a n\log(en), $$ where $C_a$ depends only on $a$. The proof uses hypercontractivity on the ambient cube to strengthen the Poincaré inequality on coordinate sections. An optimal stopping time occupation inequality for increasing sets converts the resulting local bound into a uniform bound on hitting times of large sets. An appendix by Y.~Peres improves the density dependence to $C_a=13a^{-1}\log(e/a)$, within a logarithmic factor of the conjectured $a^{-1}$ dependence.

math.PR↗

arXiv2026

Limit shapes for centrally excited walks

A centrally excited random walk on $\mathbb Z^d$ moves like simple random walk, except that its first step from each site other than the origin has a drift of fixed size toward it. Kozma (2007) conjectured that after $n$ steps the visited set approximates a ball with radius of order $n^{1/(d+1)}$. We prove this in every dimension $d\geq 2$ and show that the rescaled local times converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that, when the drift is opposite to a subgradient of a norm, the visited set approximates the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most $n^{1/6}$ times a power of $\log n$, and the exponent $1/6$ cannot be lowered. Earlier versions of this work appeared in three independent drafts by Nguyen (arXiv:2610.07724), Bou-Rabee and Peres (arxiv:2610.10162v1) and Singh (arxiv:2610.12138). The present paper brings together the results and ideas from these drafts.

math.PR↗

arXiv2026

Edit distance exponents for irrational rotations

We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \mapsto x+α$ on $\mathbb{T}$ in terms of the irrationality exponent $μ(α)$. For the binary coding determined by an interval $[0,β)$, let $d_N(x,y)$ be the edit distance between the codings of the orbits of the points $x$ and $y$ up to time $N-1$. For every $α\notin \mathbb{Q}$, almost every $β\in (0,1)$, and almost every $x,y \in \mathbb{T}$, we determine the edit-distance exponents $\displaystyle (*) \; \; \limsup_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \frac{ρ(α)}{ρ(α)+1} \; \;$ and $\; \; \displaystyle\liminf_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \min\Bigl\{\frac{1}{2},\frac{1}{ρ(α)}\Bigr\}$, where $ ρ(α):=μ(α) - 1$. We find a surprising transition at the golden mean $φ= \frac{1+\sqrt{5}}{2}$: If $ ρ(α) \le φ$, then both asymptotics in $(*)$ hold for all $β$. However, for every finite $ρ>φ$, there is an uncountable set of $α$ with $ρ(α)=ρ$ for which both edit-distance exponents are strictly smaller than the corresponding values in $(*)$ for uncountably many $β$. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of $\mathbb{T}^d$ coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is $d/(d+1)$.

math.DS↗

arXiv2026

Maximal volume of convex bodies avoiding random points and lacunary dilates

For a positive real lacunary sequence and any probability measure $μ$ with polynomial Fourier decay, we determine, for $μ$-almost every $x$, the asymptotic maximal volume of convex bodies avoiding the first $N$ dilates of $x$ modulo $\mathbb Z^d$. Among homothets of a fixed convex body, the maximal volume is asymptotic to $(\log N)/N$. When all convex bodies in the unit cube are allowed, it is asymptotic to $d(\log N)/N$. For independent uniform points in a fixed convex body $Ω\subset\mathbb R^d$, the maximal volume of an empty convex body is asymptotic to $d\operatorname{vol}(Ω)(\log N)/N$ almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.

math.NT↗

arXiv2026

Harnack inequality for $p$-harmonic functions: probabilistic and analytic approaches

We survey analytic and probabilistic approaches to the Harnack inequality for $p$-harmonic functions, $p>1$, with particular attention to the dependence of the constants on $p$ and the dimension. We explain the ideas of Moser iteration, the tug-of-war approach of Luiro, Parviainen and Saksman, and the log-gradient estimate of Kotschwar and Ni. Along the way, we give a quantitative refinement of the probabilistic argument and a direct planar proof.

math.PR↗

arXiv2026

Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality

We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $Ω\subset\mathbb R^d$, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as $p\downarrow1$. We give sufficient geometric conditions for these derivatives to be of order $(p-1)^{-1}$, and other conditions under which they decay exponentially in $1/(p-1)$. Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in $\mathbb R^{d+1}$ where the inward normal derivative is of order $\sqrt{d/(p-1)}$.

math.AP↗

arXiv2026

Survival probability in reactive gambling: monotonicity, singularity, and Hölder continuity

A gambler starts with fortune $x$ and unit bet, then aims to double the bet after every win and halve it after every loss, while the actual bet is capped by the current fortune. Each round is won independently with probability $p\in(0,1)$. We study $f(x,p)$, the probability that the gambler is never ruined. We prove that $f(x,p)>0$ exactly when $x>2$ and $p<1/2$. For fixed $x>2$, $f(x,\cdot)$ is strictly decreasing and real analytic on $(0,1/2)$, and vanishes on $[1/2,1)$: playing a more favourable game lowers the chance of surviving it. For fixed $p\in(0,1/2)$, $f(\cdot,p)$ is strictly increasing and singular continuous on $[2,\infty)$. An elementary two-step argument gives an explicit global Hölder exponent, with constant one, which is strictly larger than $\log_2(1+p)$ and tends to $\frac12\log_2(111/46)=0.635426\ldots$ as $p\uparrow1/2$; we place it in a hierarchy of computable exponents and show, by an exact finite computation, that the optimal exponent is strictly smaller than the pointwise exponent at $x=2$ once $p$ is close to $1/2$. We determine that boundary exponent, and give the complete phase diagram when wins and losses use unequal multipliers.

math.PR↗

arXiv2026

Mixing times of step-reinforced random walks

We study the mixing time of a non-Markovian process, the step-reinforced random walk (SRRW) on a finite group. This process differs from a classical random walk in that at each integer time, with probability $α$ the next step is chosen uniformly from the previous steps of the walk. We prove that the distribution of the SRRW converges to the uniform distribution exponentially fast if the walk is irreducible and aperiodic. When the step distribution is either symmetric, a class function, or has an atom at the identity, we relate the mixing time of the SRRW to the spectral gap and the mixing time of the underlying walk. For the reinforced (lazy) simple random walk on $L$-cycles, we show that the mixing time undergoes a phase transition at $α=1/2$ and the reinforcement reduces the mixing time to order $L^{1/α}$ for $α>1/2$. On the $d$-dimensional hypercube, the reinforcement slows down mixing, and the SRRW exhibits cutoff as $d\to\infty$ at time $d\log(d)/[F(α)(1-α)]$, with an $O(d)$ window for each fixed $α\in(0,1)$, where $F(\cdot)$ is a hypergeometric function.

math.PR↗

arXiv2026

Transition probabilities of step-reinforced random walks

The step-reinforced random walk (SRRW) either repeats a uniformly chosen past step or takes a fresh independent step. We consider a generalized SRRW on groups, where the selected past step is transformed by a random map. The transformations are independent of the fresh steps but may be dependent on one another. For every reinforcement parameter $α<1$, we obtain upper bounds on transition probabilities in terms of the geometry of the group. On $\mathbb{R}^d$, a non-singular step distribution yields bounds of order $n^{-d/2}$ for balls of any fixed radius, uniformly over their centers and without any moment assumption, and hence transience for $d\geq 3$. On finitely generated groups, we give bounds in terms of the isoperimetric profile. We also prove exponential decay for the elephant random walk on every nonamenable Cayley graph with a finite symmetric generating set and every memory parameter $p<1$. In particular, this answers a question of Mukherjee for Cayley trees.

math.PR↗

arXiv2026

The randomly oriented Manhattan lattice in 2D is transient

Independently orient each horizontal and vertical line of $\mathbb{Z}^2$ by a fair coin. A walker chooses one of the two lines through its current position with equal probability and takes one step in the direction of that line. Redner (1989) introduced this walk as a model of transport in an isotropic random velocity field and predicted that its root-mean-square displacement grows like $n^{2/3}$. We prove that the walk is transient almost surely. The proof is a short variational argument.

math.PR↗

arXiv2026

Once-reinforced random walk on $\mathbb{Z}^d$ has range exponent at least $d/(d+1)$

In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight $β\geq1$ after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. For sufficiently large reinforcement $β$, the expected range in the first $n$ steps was predicted by Ordemann et al. (2000) and independently by Beffara (2011) to have order $n^{d/(d+1)}$ on $\mathbb{Z}^d$. We prove the corresponding lower bound in every dimension $d\geq2$ and for every reinforcement parameter $β$.

math.PR↗

arXiv2026

Eulerian walkers on $\mathbb{Z}^2$ have range exponent $2/3$

In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by $90^\circ$ and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy (1996) introduced this as a model of self-organized criticality and conjectured that, for independent uniform initial directions, the region explored in the first $t$ steps has radius of order $t^{1/3}$. We establish this conjecture and further show that the walker visits every lattice site infinitely often, and that the region it has visited by time $t$, rescaled by $t^{1/3}$, converges to a convex body.

math.PR↗

arXiv2026

Homogenization for the $p$-Laplacian in a $d$-dimensional ball perforated along the unit sphere: the critical case $p=d$

We study a boundary value problem for the $p$-Laplacian in the perforated domain $B(0,ρ)\setminusΓ\subset \mathbb{R}^d$, where $ρ>1$ and $Γ$ is the union of many small compact cavities placed near the unit sphere. The cavities are separated at scale $\varepsilon$, asymptotically equidistributed on the sphere, and have cardinality of order $\varepsilon^{1-d}$. The cavities have diameters of order $α(\varepsilon)\varepsilon$, where $α(\varepsilon)\to0$, and their relative $p$-capacity is comparable to the relative p-capacity of a ball of the same diameter. The solution is required to equal $1$ on all cavities and $0$ on $\partial B(0,ρ)$. We focus on the critical case $p=d>1$. We identify the critical scale through the parameter $τ=\lim_{\varepsilon\downarrow0}[\varepsilon\log(1/α(\varepsilon))]^{-1}\in[0,\infty]$. Thus, $α(\varepsilon)=\exp[-(1+o(1))/(τ\varepsilon)]$ when $0<τ<\infty$. Away from the unit sphere, the solutions converge to $A_*U_ρ$, where $U_ρ(x)=\min\{1,1-\log |x|/\logρ\}$ is the radial $d$-harmonic potential of the unit ball in $B(0,ρ)$. The constant $A_*$ equals $0$ when $τ=0$, equals $1$ when $τ=\infty$, and is explicit for $0<τ<\infty$. We construct an explicit ansatz that approximates the solution for sufficiently small $\varepsilon$ in both $L^{\infty}$ and in terms of $d$-capacity.

math.AP↗

arXiv2026

Optimal lower bound for the variance of hitting times for simple random walks on graphs

We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a graph with $n$ vertices, we prove that the variance of the hitting time from a vertex $x$ to a vertex $y$, denoted $τ_y$, is at least of the order $\mathbb{E}_x(τ_y)^2 / \log n$. When the graph is a tree, we show that $n$ can be replaced by the graph's distance between vertices $x$ and $y$.

math.PR↗

arXiv2026

Meeting and coalescence times for random walks in the largest component of the Erdős-Rényi random graph

We prove that the stationary and worst-case expected meeting times of two independent continuous-time random walks on the largest component of the Erdős-Rényi random graph $G(n,p)$ have order $n$ throughout the strictly supercritical, the slightly supercritical and the critical regimes. Using these bounds along with a fine-tuned combination of comparison inequalities due to Oliveira (2012) and Kanade-Mallmann-Trenn-Sauerwald (KMS, 2023), we deduce that expected coalescence time and full voter-model consensus also have order $n$ throughout these three regimes.

math.PR↗

arXiv2026

Maximal Gaps for Dilated Lacunary Integer Sequences

Let $(a_n)_{n\ge1}\subset\mathbb{N}$ be a lacunary sequence, $a_{n+1}\ge q a_n$ for $q>1$. For $x\in\mathbb{T}$, we study the maximal empty circular gap $G_N(x)$ of the finite orbit $\{a_1x,\ldots,a_Nx\}$. We prove that, for Lebesgue-almost every $x$, \[ \frac{1}{2} \le \liminf_{N\to\infty}\frac{NG_N(x)}{\log N} \le \limsup_{N\to\infty}\frac{NG_N(x)}{\log N} \le \frac{q+1}{q-1}\,. \] If, in addition, $a_n\mid a_{n+1}$ for every $n$, then this can be improved to \[ \lim_{N\to\infty}\frac{NG_N(x)}{\log N}=1 \] for Lebesgue-almost every $x$.

math.NT↗

arXiv2026

Divisible sandpiles via random walks in random scenery

We analyze an optimal stopping problem for random walk in random scenery on general graphs, and determine when it has a finite optimum. We use this to extend a theorem of Levine, Murugan, Peres, and Ugurcan [2016]. They proved that on a vertex-transitive graph, the divisible sandpile with i.i.d. initial masses of mean $μ$ stabilizes almost surely if $μ< 1$, explodes if $μ> 1$, and explodes if $μ= 1$ with positive finite variance. Their proofs rely on conservation of mean mass under toppling. This conservation extends to unimodular random graphs, but fails on general graphs. We prove explosion for all infinite bounded-degree graphs whenever $μ\geq 1$, and stabilization for $μ<1$ provided the initial masses have finite $p$-th moment for some $p>3$. Our conditions are nearly sharp: we exhibit unbounded-degree graphs on which sandpiles with $μ> 1$ stabilize, and for every $p < 3$ we construct bounded-degree graphs on which sandpiles with~$μ< 1$ and finite $p$-th moment explode.

math.PR↗

arXiv2025

Convergence rate of $\ell^p$-relaxation on a graph to a $p$-harmonic function with given boundary values

We analyze the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Let $V = I \bigcup B$, where the set of interior vertices $I \ne \emptyset$ is disjoint from the set of boundary vertices $B \neq \emptyset$. Given $p > 1$ and an initial opinion profile $f_0: V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v_t \in I$ is selected, and the opinion there is updated to the value $f_{t}(v_t)$ that minimizes the sum $\sum_{w \sim v_t} \lvert f_t(v_t)-f_{t-1}(w) \rvert^p$ over neighbours $w$ of $v_t$. The case $p=2$ yields linear averaging dynamics, but for all $p \ne 2$ the dynamics are nonlinear. It is well known that almost surely, $f_t$ converges to the $p$-harmonic extension $h$ of $f_0 \vert_{B}$. Denote the number of steps needed to obtain $\lVert f_t - h \rVert_{\infty} \le ε$ by $τ_p(ε).$ Recently, Amir, Nazarov, and Peres~\cite{noboundarycase} analyzed the same dynamics without boundary. For individual graphs, adding boundary values can slow down the convergence considerably; indeed, when $p = 2$ the approximation time is controlled by the hitting time of the boundary by random walk, and hitting times can be much larger than mixing times, which control the convergence when $B=\emptyset$. Nevertheless, we show that for all graphs with $n$ vertices, the mean approximation time $\E[τ_p(ε)]$ is at most $n^{β_p}$ (up to logarithmic factors in $\frac{n}ε$ for $p \in [2, \infty)$, and polynomial factors in $ε^{-1}$ for $p \in (1, 2)$), where $β_p=\max\big(\frac{2p}{p-1},3\big)$. This matches the definition of $β_p$ given in \cite{noboundarycase} and answers Question 6.2 in that paper. The exponent $β_p$ is optimal in both settings. We also prove sharp bounds for $n$-vertex graphs with given average degree, that are technically more challenging.

math.PR↗