arXiv subjects

Michael Damron

Publications and source records attributed to Michael Damron.

At least 37 records · Page 2Linked to original sources

arXiv2018

Random growth models: shape and convergence rate

Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the typical models and the basic analytical questions and properties, like existence of asymptotic shapes, fluctuations of infection times, and relations to particle systems. We then specialize to models built on percolation (first-passage percolation and last-passage percolation) and give a self-contained treatment of the shape theorem, the subadditive ergodic theorem, and conjectured and proven properties of asymptotic shapes. We finish by discussing the rate of convergence to the limit shape, along with definitions of scaling exponents and a sketch of the proof of the KPZ scaling relation.

math.PR↗

arXiv2018

Coarsening model on $\mathbb{Z}^d$ with biased zero-energy flips and an exponential large deviation bound for ASEP

We study the coarsening model (zero-temperature Ising Glauber dynamics) on $\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a Markov process on the state space $\{-1,+1\}^{\mathbb{Z}^d}$ of "spin configurations" in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$ with probability $q \in [0,1]$ and to $-1$ with probability $1-q$. The initial state of this Markov chain is distributed according to a product measure with probability $p$ for a spin to be $+1$. In this paper, we show that for any given $p>0$, there exist $q$ close enough to 1 such that a.s. every spin has a limit of $+1$. This is of particular interest for small values of $p$, for which it is known that if $q=1/2$, a.s. all spins have a limit of $-1$. For dimension $d=2$, we also obtain near-exponential convergence rates for $q$ sufficiently large, and for general $d$, we obtain stretched exponential rates independent of $d$. Two important ingredients in our proofs are refinements of block arguments of Fontes-Schonmann-Sidoravicius and a novel exponential large deviation bound for the Asymmetric Simple Exclusion Process.

math.PR↗

arXiv2017

The size of the boundary in first-passage percolation

First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_e)$ on the nearest-neighbor edges of $\mathbb{Z}^d$. An initial infection occupies the origin and spreads along the edges, taking time $t_e$ to cross the edge $e$. In this paper, we study the size of the boundary of the infected ("wet") region at time $t$, $B(t)$. It is known that $B(t)$ grows linearly, so its boundary $\partial B(t)$ has size between $ct^{d-1}$ and $Ct^d$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\partial B(t)$ has size of order $t^{d-1+α}$ for some $α>0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ (edges touching the unbounded component of the complement of $B(t)$) is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\#\partial B(t) \leq (\log t)^C t^{d-1}$ for all large $t.$

math.PR↗

arXiv2017

Arm events in two-dimensional invasion percolation

We compare the probabilities of arm events in two-dimensional invasion percolation to those in critical percolation. Arm events are defined by the existence of a prescribed color sequence of invaded and non-invaded connections from the origin to distance n. We find that, for sequences of a particular form, arm probabilities in invasion percolation and critical percolation are comparable, uniformly in n, while they differ by a power of n for all others. A corollary of our results is the existence, on the triangular lattice, of arm exponents for invasion percolation, for any color sequence with at least two open (invaded) entries.

math.PR↗

arXiv2017

Strict inequality for the chemical distance exponent in two-dimensional critical percolation

We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-δ}π_3(n)$, for some $δ>0$, where $π_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain a strict upper bound for the exponent of $4/3$. The proof builds on the strategy developed in our previous paper, but with a new iterative scheme, and a new large deviation inequality for events in annuli conditional on arm events, which may be of independent interest.

math.PR↗

arXiv2016

Entropy reduction in Euclidean first-passage percolation

The Euclidean first-passage percolation (FPP) model of Howard and Newman is a rotationally invariant model of FPP which is built on a graph whose vertices are the points of homogeneous Poisson point process. It was shown that one has (stretched) exponential concentration of the passage time $T_n$ from $0$ to $n\mathbf{e}_1$ about its mean on scale $\sqrt{n}$, and this was used to show the bound $μn \leq \mathbb{E}T_n \leq μn + C\sqrt{n} (\log n)^a$ for $a,C>0$ on the discrepancy between the expected passage time and its deterministic approximation $μ= \lim_n \frac{\mathbb{E}T_n}{n}$. In this paper, we introduce an inductive entropy reduction technique that gives the stronger upper bound $\mathbb{E}T_n \leq μn + C_kψ(n) \log^{(k)}n$, where $ψ(n)$ is a general scale of concentration and $\log^{(k)}$ is the $k$-th iterate of $\log$. This gives evidence that the inequality $\mathbb{E}T_n - μn \leq C\sqrt{\mathrm{Var}~T_n}$ may hold.

math.PR↗

arXiv2016

Superlinearity of geodesic length in 2$D$ critical first-passage percolation

First-passage percolation is the study of the metric space $(\mathbb{Z}^d,T)$, where $T$ is a random metric defined as the weighted graph metric using random edge-weights $(t_e)_{e\in \mathcal{E}^d}$ assigned to the nearest-neighbor edges $\mathcal{E}^d$ of the $d$-dimensional cubic lattice. We study the so-called critical case in two dimensions, in which $\mathbb{P}(t_e=0)=p_c$, where $p_c$ is the threshold for two-dimensional bond percolation. In contrast to the standard case $( 1$ such that with probability at least $1-e^{-\|x\|_1^c}$, the minimal length geodesic from $0$ to $x$ has at least $\|x\|_1^s$ number of edges. Our proofs combine recent ideas to bound $T$ for general critical distributions, and modifications of techniques of Aizenman-Burchard to estimate the Hausdorff dimension of random curves.

math.PR↗

arXiv2016

50 years of first passage percolation

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2+log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.

math.PR↗

arXiv2016

Bigeodesics in first-passage percolation

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z^2 and consider the weighted graph metric. A distance-minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the '90s, Licea-Newman showed that, under a curvature assumption on the "asymptotic shape", all infinite geodesics have an asymptotic direction, and there is a full-measure set D of [0, 2 pi) such that for any theta in D, there are no bigeodesics with one end directed in direction theta. In this paper, we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable. This rules out existence of ground state pairs for the related disordered ferromagnet whose interface has a deterministic direction. Furthermore, it resolves the Benjamini-Kalai-Schramm "midpoint problem" under the extra assumption that the limit shape boundary is differentiable.

math.PR↗

arXiv2016

Rate of convergence in first-passage percolation under low moments

We consider first-passage percolation on the $d$ dimensional cubic lattice for $d \geq 2$; that is, we assign independently to each edge $e$ a nonnegative random weight $t_e$ with a common distribution and consider the induced random graph distance (the passage time), $T(x,y)$. It is known that for each $x \in \mathbb{Z}^d$, $μ(x) = \lim_n T(0,nx)/n$ exists and that $0 \leq \mathbb{E}T(0,x) - μ(x) \leq C\|x\|_1^{1/2}\log \|x\|_1$ under the condition $\mathbb{E}e^{αt_e}<\infty$ for some $α>0$. By combining tools from concentration of measure with Alexander's methods, we show how such bounds can be extended to $t_e$'s with distributions that have only low moments. For such edge-weights, we obtain an improved bound $C (\|x\|_1 \log \|x\|_1)^{1/2}$ and bounds on the rate of convergence to the limit shape.

math.PR↗

arXiv2016

Recent work on chemical distance in critical percolation

In this note, we describe some of the progress recently made on questions regarding the chemical distance in two-dimensional critical percolation by the author, J. Hanson, and P. Sosoe [6, 7]. It is expected that the distance between points in critical percolation clusters scales as $\|\cdot \|^{1+s}$, where $\|\cdot \|$ is the Euclidean distance and $s>0$. First, we review previous work of Aizenman-Burchard and Morrow-Zhang, which together establish a version of $0 < s \leq 1/3$. The main results of our work are in the direction of proving upper bounds on $s$, answering in [6] a question from '93 of Kesten-Zhang on the ratio of the length of the shortest crossing of a box to the length of the lowest crossing of a box. The paper [7] provides a quantitative version of the result of [6], along with bounds on point-to-point and point-to-set distances.

math.PR↗

arXiv2016

On the chemical distance in critical percolation II

We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in $\mathbb{Z}^2$ between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.

math.PR↗

arXiv2015

On the number of ergodic measures for minimal shifts with eventually constant complexity growth

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length n relates to the one for words of length n+1 for each n = 1,2,3...

math.DS↗

arXiv2015

Asymptotics for $2D$ Critical First Passage Percolation

We consider first-passage percolation on $\mathbb{Z}^2$ with i.i.d. weights, whose distribution function satisfies $F(0) = p_c = 1/2$. This is sometimes known as the "critical case" because large clusters of zero-weight edges force passage times to grow at most logarithmically, giving zero time constant. Denote $T(\mathbf{0}, \partial B(n))$ as the passage time from the origin to the boundary of the box $[-n,n] \times [-n,n]$. We characterize the limit behavior of $T(\mathbf{0}, \partial B(n))$ by conditions on the distribution function $F$. We also give exact conditions under which $T(\mathbf{0}, \partial B(n))$ will have uniformly bounded mean or variance. These results answer several questions of Kesten and Zhang from the '90s and, in particular, disprove a conjecture of Zhang from '99. In the case when both the mean and the variance go to infinity as $n \to \infty$, we prove a CLT under a minimal moment assumption. The main tool involves a new relation between first-passage percolation and invasion percolation: up to a constant factor, the passage time in critical first-passage percolation has the same first-order behavior as the passage time of an optimal path constrained to lie in an embedded invasion cluster.

math.PR↗

arXiv2015

On the chemical distance in critical percolation

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

math.PR↗

arXiv2015

Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^2$

It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this question for first-passage percolation on a wide class of subgraphs of $\mathbb {Z}^2$: those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing $x_n$ for the sequence of boundary vertices, we show that the sequence of geodesics from any point to $x_n$ has an almost sure limit assuming only existence of finite geodesics. For all passage-time configurations, we show existence of a limiting Busemann function. Specializing to the case of the half-plane, we prove that the limiting geodesic graph has one topological end; that is, all its infinite geodesics coalesce, and there are no backward infinite paths. To do this, we prove in the Appendix existence of geodesics for all product measures in our domains and remove the moment assumption of the Wehr-Woo theorem on absence of bigeodesics in the half-plane.

math.PR↗

arXiv2014

Subdiffusive concentration in first-passage percolation

We prove exponential concentration in i.i.d. first-passage percolation in $Z^d$ for all $d \geq 2$ and general edge-weights $(t_e)$. Precisely, under an exponential moment assumption $E e^{αt_e}< \infty$ for some $α>0$) on the edge-weight distribution, we prove the inequality $$ P(|T(0,x)-E T(0,x)| \geq λ\sqrt{\frac{|x|}{log |x|}}) \leq ce^{-c' λ}, |x|>1 $$ for the point-to-point passage time $T(0,x)$. Under a weaker assumption $E t_e^2(\log t_e)_+< \infty$ we show a corresponding inequality for the lower-tail of the distribution of $T(0,x)$. These results extend work of Benaim-Rossignol to general distributions.

math.PR↗

arXiv2014

Sublinear variance in first-passage percolation for general distributions

We prove that the variance of the passage time from the origin to a point x in first-passage percolation on Z^d is sublinear in the distance to x when d \geq 2, obeying the bound Cx/(log x), under minimal assumptions on the edge-weight distribution. The proof applies equally to absolutely continuous, discrete and singular continuous distributions and mixtures thereof, and requires only 2+log moments. The main result extends work of Benjamini-Kalai-Schramm and Benaim-Rossignol.

math.PR↗