Searcharxiv⌕ Search

arXiv subjects

Vladas Sidoravicius

Publications and source records attributed to Vladas Sidoravicius.

At least 19 recordsLinked to original sources
arXiv2026

Rotationally invariant first passage percolation: Breaking the $n/\log n$ variance barrier

For first passage percolation (FPP) on Euclidean lattices $\mathbb{Z}^d$ with $d\ge 2$, it is expected that the variance of the first passage time between two points grows sublinearly in the distance with a universal exponent strictly smaller than $1$. Following Kesten's $O(n)$ upper bound (Ann. Appl. Probab., 1993) on the variance, Benjamini, Kalai and Schramm (Ann. Probab., 2003) used hypercontractivity to obtain an improvement of a factor of $\log n$ when passage times take two values with equal probability. This was later extended to more general classes of passage time distributions. However, unlike in exactly solvable planar models in last passage percolation where the variance is known to be $Θ(n^{2/3})$, the best known upper bound for the variance of passage times has remained $O(n/\log n)$ in all non-trivial variants of FPP. For a class of rotationally invariant Riemannian FPP on the plane, we show that the variance is $O(n^{1-\varepsilon})$ for some $\varepsilon>0$. Our argument uses fluctuation estimates for passage times and geodesics derived in Basu, Sidoravicius and Sly (2023) together with a multi-scale argument to establish that the geodesic exhibits disorder chaos, i.e., upon resampling a small fraction of the underlying randomness, the updated geodesic has on average a small overlap with the original one; this, established at a large number of scales, leads to a polynomial improvement of the variance bound.

math.PR↗
arXiv2023

Rotationally invariant first passage percolation: Concentration and scaling relations

For rotationally invariant first passage percolation (FPP) on the plane, we use a multi-scale argument to prove stretched exponential concentration of the first passage times at the scale of the standard deviation. Our results are proved under hypotheses which can be verified for many standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the Howard-Newman model. This is the first such tight concentration result known for any model that is not exactly solvable. As a consequence, we prove a version of the so called KPZ relation between the passage time fluctuations and the transversal fluctuations of geodesics as well as up to constant upper and lower bounds for the non-random fluctuations in these models. Similar results have previously been known conditionally under unproven hypotheses, but our results are the first ones that apply to some specific FPP models. Our arguments are expected to be useful in proving a number of other estimates which were hitherto only known conditionally or for exactly solvable models.

math.PR↗
arXiv2022

Dependent Percolation on $\mathbb{Z}^2$

We consider a dependent percolation model on the square lattice $\mathbb{Z}^2$. The range of dependence is infinite in vertical and horizontal directions. In this context, we prove the existence of a phase transition. The proof exploits a multi-scale renormalization argument that is defined once the environment configuration is suitably good and, which, together with the main estimate for the induction step, comes from Kesten, Sidoravicius and Vares (To appear in {\em Electronic Journal of Probability}, (2022)). This work was inspired by de Lima (Ph.D.Thesis, \emph{Informes de Matemática. IMPA}, Série C-26/2004) where the simpler case of a deterministic environment was considered. It has various applications, including an alternative proof for the phase transition on the two dimensional random stretched lattice proved by Hoffman ({\em Comm. Math. Phys.} {\bf 254}, 1-22 (2005)).

math.PR↗
arXiv2021

Three-speed ballistic annihilation: phase transition and universality

We consider ballistic annihilation, a model for chemical reactions first introduced in the 1980's physics literature. In this particle system, initial locations are given by a renewal process on the line, motions are ballistic - i.e. each particle is assigned an i.i.d. constant velocity - and collisions between pairs of particles result in mutual annihilation. We focus on the case when the velocities are symmetrically distributed among three values, i.e. particles either remain static (with given probability~$p$) or move at constant velocity uniformly chosen among $\pm1$. We establish that this model goes through a phase transition at $p_c=1/4$ between a subcritical regime where every particle eventually annihilates, and a supercritical regime where a positive density of static particles is never hit, confirming 1990s predictions of Droz et al. for the particular case of a Poisson process. Our result encompasses cases where triple collisions can happen; these are resolved by annihilation of one static and one randomly chosen moving particle. Our arguments, of combinatorial nature, show that, although the model is not completely solvable, certain large scale features can be explicitly computed, and are universal, i.e. insensitive to the distribution of the initial point process. In particular, in the critical and subcritical regimes, the asymptotics of the time decay of the densities of each type of particle is universal (among exponentially integrable interdistance distributions) and, in the supercritical regime, the distribution of the ``skyline'' process, i.e. the process restricted to the last particles to ever visit a location, has a universal description. We also prove that an alternative model introduced by Burdinski, Gupta and Junge does not share the same universality as our model, and find numerical bounds on its critical probability.

math.PR↗
arXiv2020

One-dimensional Multi-particle DLA -- a PDE approach

In the present note we analyze the one-dimensional multi-particle diffusion limited aggregation (MDLA) model: the initial number of particles at each positive integer site has Poisson distribution with mean $μ$, independently of all other sites. Particles perform independent continuous-time simple symmetric random walks until they come to the site neighbouring the sticky aggregate, which initially consists only of the origin. If a particle tries to jump on the aggregate, the size of the aggregate increases by one, i.e., its rightmost point moves to the right by one unit. All particles which are present at the site neighbouring the aggregate at the moment when the aggregate advances, are immediately deleted. The $d-$dimensional MDLA model, $d \geq 1$, was introduced in 1980 by Rosenstock and Marquardt, and studied numerically by Voss (1984). The one dimensional model exhibits a phase transition for the rate of growth of the aggregate: it was proven by Kesten and Sidoravicius (2008) that if $μ<1$ then the size $R(t)$ of the aggregate grows like $\sqrt{t}$ and Sly (2016+) proved that if $μ>1$ then $R(t)$ grows linearly. In this note we give heuristic predictions about the constant $c(μ)$ for which $R(t)\approx c(μ)\sqrt{t}$ in the subcritical case $μ<1$, $R(t)\approx c(1+\varepsilon)t$ in the barely supercritical case $μ=1+\varepsilon$ and $R(t) \approx c(1) t^{2/3}$ in the critical case $μ=1$. We compare our predictions with new computer simulation results of the 1-dimensional multi-particle DLA model.

math-ph↗
arXiv2020

Averaging Principle and Shape Theorem for a Growth Model with Memory

We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a "limiting shape". We prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain.

math.PR↗
arXiv2020

Limit set of branching random walks on hyperbolic groups

Let $Γ$ be a nonelementary hyperbolic group with a word metric $d$ and $\partialΓ$ its hyperbolic boundary equipped with a visual metric $d_a$ for some parameter $a>1$. Fix a superexponential symmetric probability $μ$ on $Γ$ whose support generates $Γ$ as a semigroup, and denote by $ρ$ the spectral radius of the random walk $Y$ on $Γ$ with step distribution $μ$. Let $ν$ be a probability on $\{1,\, 2, \, 3, \, \ldots\}$ with mean $λ=\sum\limits_{k=1}^\infty kν(k)<\infty$. Let $\mathrm{BRW}(Γ, \, ν, \, μ)$ be the branching random walk on $Γ$ with offspring distribution $ν$ and base motion $Y$ and $H(λ)$ the volume growth rate for the trace of $\mathrm{BRW}(Γ, \, ν, \, μ)$. We prove for $λ\in [1, \, ρ^{-1})$ that the Hausdorff dimension of the limit set $Λ$, which is the random subset of $(\partial Γ, \, d_a)$ consisting of all accumulation points of the trace of $\mathrm{BRW}(Γ, \, ν, \, μ)$, is given by $\log_a H(λ)$. Furthermore, we prove that $H(λ)$ is almost surely a deterministic, strictly increasing and continuous function of $λ\in [1, \, ρ^{-1}]$, is bounded by the square root of the volume growth rate of $Γ$, and has critical exponent $1/2$ at $ρ^{-1}$ in the sense that \[ H(ρ^{-1}) - H(λ) \sim C \sqrt{ρ^{-1} - λ} \quad \text{as } λ\uparrow ρ^{-1} \] for some positive constant $C$. We conjecture that the Hausdorff dimension of $Λ$ in the critical case $λ=ρ^{-1}$ is $\log_aH(ρ^{-1})$ almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric $d$ defined by the standard generating set.

math.PR↗
arXiv2020

Bernoulli Hyperplane Percolation

We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.

math.PR↗
arXiv2020

Random Memory Walk

We present a simple model of a random walk with partial memory, which we call the \emph{random memory walk}. We introduce this model motivated by the belief that it mimics the behavior of the once-reinforced random walk in high dimensions and with small reinforcement. We establish the transience of the random memory walk in dimensions three and higher, and show that its scaling limit is a Brownian motion.

math.PR↗
arXiv2020

The Constrained-degree percolation model

In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open at time $U_e$, it succeeds if both its end-vertices would have degrees at most $k-1$. We prove a phase transition theorem for this model on the square lattice $\mathbb{L}^2$, as well as on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase.

math.PR↗
arXiv2019

Percolation of finite clusters and shielded paths

In independent bond percolation on $\mathbb{Z}^d$ with parameter $p$, if one removes the vertices of the infinite cluster (and incident edges), for which values of $p$ does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for $d \geq 19$, the set of such $p$ contains values strictly larger than the percolation threshold $p_c$. With the work of Fitzner-van der Hofstad, this has been reduced to $d \geq 11$. We improve this result by showing that for $d \geq 10$ and some $p>p_c$, there are infinite paths consisting of "shielded" vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of $p_c$, this bound can be reduced to $d \geq 7$. Our methods are elementary and do not require the triangle condition.

math.PR↗
arXiv2019

Multi-Particle Diffusion Limited Aggregation

We consider a stochastic aggregation model on Z^d. Start with particles located at the vertices of the lattice, initially distributed according to the product Bernoulli measure with parameter μ. In addition, there is an aggregate, which initially consists of the origin. Non-aggregated particles move as continuous time simple random walks obeying the exclusion rule, whereas aggregated particles do not move. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. This evolution is referred to as multi-particle diffusion limited aggregation. Our main result states that if on d>1 the initial density of particles is large enough, then with positive probability the aggregate has linearly growing arms, i.e. if F(t) denotes the point of the aggregate furthest away from the origin at time t>0, then there exists a constant c>0 so that |F(t)|>ct, for all t eventually. The key conceptual element of our analysis is the introduction and study of a new growth process. Consider a first passage percolation process, called type 1, starting from the origin. Whenever type 1 is about to occupy a new vertex, with positive probability, instead of doing it, it gives rise to another first passage percolation process, called type 2, which starts to spread from that vertex. Each vertex gets occupied only by the process that arrives to it first. This process may have three phases: an extinction phase, where type 1 gets eventually surrounded by type 2 clusters, a coexistence phase, where infinite clusters of both types emerge, and a strong survival phase, where type 1 produces an infinite cluster that successfully surrounds all type 2 clusters. Understanding the behavior of this process in its various phases is of mathematical interest on its own right. We establish the existence of a strong survival phase, and use this to show our main result.

math.PR↗
arXiv2019

Universality and Sharpness in Absorbing-State Phase Transitions

We consider the Activated Random Walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice.

math.PR↗
arXiv2018

The branching-ruin number and the critical parameter of once-reinforced random walk on trees

The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. computable) criterion characterizing the recurrence/transience of a larger class of self-interacting walks on trees, providing the complete picture for their phase transition.

math.PR↗
arXiv2018

Stability of the Greedy Algorithm on the Circle

We consider a single-server system with service stations in each point of the circle. Customers arrive after exponential times at uniformly-distributed locations. The server moves at finite speed and adopts a greedy routing mechanism. It was conjectured by Coffman and Gilbert in~1987 that the service rate exceeding the arrival rate is a sufficient condition for the system to be positive recurrent, for any value of the speed. In this paper we show that the conjecture holds true.

math.PR↗
arXiv2018

The ballistic annihilation threshold is 1/4

We consider a system of annihilating particles where particles start from the points of a Poisson process on the line, move at constant i.i.d. speeds symmetrically distributed in {-1,0,+1} and annihilate upon collision. We prove that particles with speed 0 vanish almost surely if and only if their initial density is smaller than or equal to 1/4, and give an explicit formula for the probability of survival of a stationary particle, which is in accordance with the predictions of [Droz et al. 1995]. The present proof relies essentially on an identity proved in a recent paper by J. Haslegrave.

math.PR↗