arXiv subjects

Michael Damron

Publications and source records attributed to Michael Damron.

At least 19 recordsLinked to original sources

arXiv2025

Asymptotics for first-passage percolation on logarithmic subgraphs of $\mathbb{Z}^2$

For $a>0$ and $b \geq 0$, let $\mathbb{G}_{a,b}$ be the subgraph of $\mathbb{Z}^2$ induced by the vertices between the first coordinate axis and the graph of the function $f = f_{a,b}(u) = a \log (1+u) + b \log(1+\log(1+u))$, $u \geq 0$. It is known that for $a>0$, the critical value for Bernoulli percolation on $\mathbb{G}_f = \mathbb{G}_{a,b}$ is strictly between $1/2$ and $1$, and that if $b>2a$ then the percolation phase transition is discontinuous. We study first-passage percolation (FPP) on $\mathbb{G}_{a,b}$ with i.i.d. edge-weights $(\tau_e)$ satisfying $p = \mathbb{P}(\tau_e=0) \in [1/2,1)$ and the "gap condition" $\mathbb{P}(\tau_e \leq \delta) = p$ for some $\delta>0$. We find the rate of growth of the expected passage time in $\mathbb{G}_f$ from the origin to the line $x=n$, and show that, while when $p=1/2$ it is of order $n/(a \log n)$, when $p>1/2$ it can be of order (a) $n^{c_1}/(\log n)^{c_2}$, (b) $(\log n)^{c_3}$, (c) $\log \log n$, or (d) constant, depending on the relationship between $a,b,$ and $p$. For more general functions $f$, we prove a central limit theorem for the passage time and show that its variance grows at the same rate as the mean. As a consequence of our methods, we improve the percolation transition result by showing that the phase transition on $\mathbb{G}_{a,b}$ is discontinuous if and only if $b > a$, and improve "sponge crossing dimensions" asymptotics from the '80s on subcritical percolation crossing probabilities for tall thin rectangles.

math.PR↗

arXiv2023

Exceptional behavior in critical first-passage percolation and random sums

We study first-passage percolation (FPP) on the square lattice. The model is defined using i.i.d. nonnegative random edge-weights $(t_e)$ associated to the nearest neighbor edges of $\mathbb{Z}^2$. The passage time between vertices $x$ and $y$, $T(x,y)$, is the minimal total weight of any lattice path from $x$ to $y$. The growth rate of $T(x,y)$ depends on the value of $F(0) = \mathbb{P}(t_e=0)$: if $F(0) < 1/2$ then $T(x,y)$ grows linearly in $|x-y|$, but if $F(0) > 1/2$ then it is stochastically bounded. In the critical case, where $F(0) = 1/2$, $T(x,y)$ can be bounded or unbounded depending on the behavior of the distribution function $F$ of $t_e$ near 0. In this paper, we consider the critical case in which $T(x,y)$ is unbounded and prove the existence of an incipient infinite cluster (IIC) type measure, constructed by conditioning the environment on the event that the passage time from $0$ to a far distance remains bounded. This IIC measure is a natural candidate for the distribution of the weights at a typical exceptional time in dynamical FPP. A major part of the analysis involves characterizing the limiting behavior of independent nonnegative random variables conditioned to have small sum. We give conditions on random variables that ensure that such limits are trivial, and several examples that exhibit nontrivial limits.

math.PR↗

arXiv2022

On the number and size of holes in the growing ball of first-passage percolation

First-passage percolation is a random growth model defined on $\mathbb{Z}^d$ using i.i.d. nonnegative weights $(\tau_e)$ on the edges. Letting $T(x,y)$ be the distance between vertices $x$ and $y$ induced by the weights, we study the random ball of radius $t$ centered at the origin, $B(t) = \{x \in \mathbb{Z}^d : T(0,x) \leq t\}$. It is known that for all such $\tau_e$, the number of vertices (volume) of $B(t)$ is at least order $t^d$, and under mild conditions on $\tau_e$, this volume grows like a deterministic constant times $t^d$. Defining a hole in $B(t)$ to be a bounded component of the complement $B(t)^c$, we prove that if $\tau_e$ is not deterministic, then a.s., for all large $t$, $B(t)$ has at least $ct^{d-1}$ many holes, and the maximal volume of any hole is at least $c\log t$. Conditionally on the (unproved) uniform curvature assumption, we prove that a.s., for all large $t$, the number of holes is at most $(\log t)^C t^{d-1}$, and for $d=2$, no hole in $B(t)$ has volume larger than $(\log t)^C$. Without curvature, we show that no hole has volume larger than $Ct \log t$.

math.PR↗

arXiv2022

Fluctuation bounds for first-passage percolation on the square, tube, and torus

In first-passage percolation, one assigns i.i.d. nonnegative weights $(t_e)$ to the edges of $\mathbb{Z}^d$ and studies the induced distance (passage time) $T(x,y)$ between vertices $x$ and $y$. It is known that for $d=2$, the fluctuations of $T(x,y)$ are at least order $\sqrt{\log |x-y|}$ under mild assumptions on $t_e$. We study the question of fluctuation lower bounds for $T_n$, the minimal passage time between two opposite sides of an $n$ by $n$ square. The main result is that, under a curvature assumption, this quantity has fluctuations at least of order $n^{1/8-ε}$ for any $ε>0$ when the $t_e$ are exponentially distributed. As previous arguments to bound the fluctuations of $T(x,y)$ only give a constant lower bound for those of $T_n$ (even assuming curvature), a different argument, representing $T_n$ as a minimum of cylinder passage times, and deriving more detailed information about the distribution of cylinder times using the Markov property, is developed. As a corollary, we obtain the first polynomial lower bounds on higher central moments of the discrete torus passage time, under the same curvature assumption.

math.PR↗

arXiv2021

Transitions for exceptional times in dynamical first-passage percolation

In first-passage percolation (FPP), we let $(τ_v)$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $τ_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if $\sum F^{-1}(1/2+1/2^k) = \infty$, then a.s. there are exceptional times at which the weight grows atypically, but if $\sum k^{7/8} F^{-1}(1/2+1/2^k) <\infty$, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP.

math.PR↗

arXiv2020

Estimates for the empirical distribution along a geodesic in first-passage percolation

In first-passage percolation, we assign i.i.d.~nonnegative weights $(t_e)$ to the nearest-neighbor edges of $\mathbb{Z}^d$ and study the induced pseudometric $T = T(x,y)$. In this paper, we focus on geodesics, or optimal paths for $T$, and estimate the empirical distribution of weights along them. We prove an upper bound for the expected number of edges with weight $\geq M$ in the union of all geodesics from $0$ to $x$ of the form $q(M) \mathbb{P}(t_e \geq M)|x|$, where $q(M) \leq e^{-cM}$. This shows that the tail of the expected empirical distribution along a geodesic is lighter than that of the original weight distribution by an exponential factor. We also give a lower bound for the expected minimal number of edges with weight $\geq M$ in any geodesic from $0$ to $x$ in terms of $\mathbb{P}(t_e \geq M)$ and $\mathbb{P}(t_e \in [M,2M])$. For example, these two imply that if $t_e$ has a power law tail of the form $\mathbb{P}(t_e \geq M) \sim M^{-\alpha}$, then the tail of the expected empirical distribution asymptotically lies between $e^{-CM \log M}$ and $e^{-cM}$. We also provide estimates for the expected number of edges in a geodesic with weight in a set $A$ for (a) arbitrary $A$, (b) $A$ an interval separated from the infimum of the support of $t_e$ and (c) $A=[0,a]$ for some $a \geq 0$.

math.PR↗

arXiv2020

Stretched exponential decay for subcritical parking times on $\mathbb{Z}^d$

In the parking model on $\mathbb{Z}^d$, each vertex is initially occupied by a car (with probability $p$) or by a vacant parking spot (with probability $1-p$). Cars perform independent random walks and when they enter a vacant spot, they park there, thereby rendering the spot occupied. Cars visiting occupied spots simply keep driving (continuing their random walk). It is known that $p=1/2$ is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when $p<1/2$, and by infinitely many distinct cars when $p\geq 1/2$. Furthermore, any given car a.s. eventually parks for $p \leq 1/2$ and with positive probability does not park for $p > 1/2$. We study the subcritical phase and prove that the tail of the parking time $τ$ of the car initially at the origin obeys the bounds \[ \exp\left( - C_1 t^{\frac{d}{d+2}}\right) \leq \mathbb{P}_p(τ> t) \leq \exp\left( - c_2 t^{\frac{d}{d+2}}\right) \] for $p>0$ sufficiently small. For $d=1$, we prove these inequalities for all $p \in [0,1/2)$. This result presents an asymmetry with the supercritical phase ($p>1/2$), where methods of Bramson--Lebowitz imply that for $d=1$ the corresponding tail of the parking time of the parking spot of the origin decays like $e^{-c\sqrt{t}}$. Our exponent $d/(d+2)$ also differs from those previously obtained in the case of moving obstacles.

math.PR↗

arXiv2020

Random nearest neighbor graphs: the translation invariant case

If $(ω(e))$ is a family of random variables (weights) assigned to the edges of $\mathbb{Z}^d$, the nearest neighbor graph is the directed graph induced by all edges $\langle x,y \rangle$ such that $ω(\{x,y\})$ is minimal among all neighbors $y$ of $x$. That is, each vertex points to its closest neighbor, if the weights are viewed as edge-lengths. Nanda-Newman introduced nearest neighbor graphs when the weights are i.i.d. and continuously distributed and proved that a.s., all components of the undirected version of the graph are finite. We study the case of translation invariant, distinct weights, and prove that nearest neighbor graphs do not contain doubly-infinite directed paths. In contrast to the i.i.d. case, we show that in this stationary case, the graphs can contain either one or two infinite components (but not more) in dimension two, and $k$ infinite components for any $k \in [1,\infty]$ in dimension $\geq 3$. The latter constructions use a general procedure to exhibit a certain class of directed graphs as nearest neighbor graphs with distinct weights, and thereby characterize all translation invariant nearest neighbor graphs. We also discuss relations to geodesic graphs from first-passage percolation and implications for the coalescing walk model of Chaika-Krishnan.

math.PR↗

arXiv2020

Absence of backward infinite paths for first-passage percolation in arbitrary dimension

In first-passage percolation (FPP), one places nonnegative random variables (weights) $(t_e)$ on the edges of a graph and studies the induced weighted graph metric. We consider FPP on $\mathbb{Z}^d$ for $d \geq 2$ and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of Furstenberg and most strongly supported for $d=2$ that for continuously distributed i.i.d. weights, there a.s. are no bigeodesics. We provide the first progress on this question in general dimensions under no unproven assumptions. Our main result is that geodesic graphs, introduced in a previous paper of two of the authors, constructed in any deterministic direction a.s. do not contain doubly-infinite paths. As a consequence, one can construct random graphs of subsequential limits of point-to-hyperplane geodesics which contain no bigeodesics. This gives evidence that bigeodesics, if they exist, cannot be constructed in a translation-invariant manner as limits of point-to-hyperplane geodesics.

math.PR↗

arXiv2020

The number of ergodic measures for transitive subshifts under the regular bispecial condition

If $\mathcal{A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal{A}^{\mathbb{N}}$ of sequences with entries in $\mathcal{A}$, along with the left shift operator $S$. Closed $S$-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition ("regular bispecial condition") on the possible extensions of words in the associated language. Our main result shows that under this condition, the subshift can support at most $\frac{K+1}{2}$ ergodic measures, where $K$ is the limiting value of $p(n+1)-p(n)$, and $p$ is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from `84, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.

math.DS↗

arXiv2019

Non-Optimality of Invaded Geodesics in 2d Critical First-Passage Percolation

We study the critical case of first-passage percolation in two dimensions. Letting $(t_e)$ be i.i.d. nonnegative weights assigned to the edges of $\mathbb{Z}^2$ with $\mathbb{P}(t_e=0)=1/2$, consider the induced pseudometric (passage time) $T(x,y)$ for vertices $x,y$. It was shown in [2] that the growth of the sequence $\mathbb{E}T(0,\partial B(n))$ (where $B(n) = [-n,n]^2$) has the same order (up to a constant factor) as the sequence $\mathbb{E}T^{\text{inv}}(0,\partial B(n))$. This second passage time is the minimal total weight of any path from 0 to $\partial B(n)$ that resides in a certain embedded invasion percolation cluster. In this paper, we show that this constant factor cannot be taken to be 1. That is, there exists $c>0$ such that for all $n$, \[ \mathbb{E}T^{\text{inv}}(0,\partial B(n)) \geq (1+c) \mathbb{E}T(0,\partial B(n)). \] This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structure.

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

Lower bounds for fluctuations in first-passage percolation for general distributions

In first-passage percolation (FPP), one assigns i.i.d.~weights to the edges of the cubic lattice $\mathbb{Z}^d$ and analyzes the induced weighted graph metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a primary question in the model is: what is the order of the fluctuations of $T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of $x$ to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order $\log \|x\|$. This result was found in the '90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is with high probability not contained in an interval of size $o(\log \|x\|)^{1/2}$, and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of "hi-mode" (large).

math.PR↗

arXiv2019

The acceptance profile of invasion percolation at $p_c$ in two dimensions

Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of $\mathbb{Z}^d$, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the `acceptance profile' of the invasion: for a given $p \in [0,1]$, it is the ratio of the expected number of invaded edges until time $n$ with weight in $[p,p+\text{d}p]$ to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile $a_n(p)$ converges to one for $p p_c$. In this paper, we consider $a_n(p)$ at the critical point $p=p_c$ in two dimensions and show that it is bounded away from zero and one as $n \to \infty$.

math.PR↗

arXiv2019

Universality of the time constant for $2D$ critical first-passage percolation

We consider first-passage percolation (FPP) on the triangular lattice with vertex weights $(t_v)$ whose common distribution function $F$ satisfies $F(0)=1/2$. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by $T(0,\partial B(n))$ the first-passage time from $0$ to $\{x : \|x\|_\infty = n\}$, we show existence of the "time constant'' and find its exact value to be \[ \lim_{n \to \infty} \frac{T(0,\partial B(n))}{\log n} = \frac{I}{2\sqrt{3}π} \text{ almost surely}, \] where $I = \inf\{x > 0 : F(x) > 1/2\}$ and $F$ is any critical distribution for $t_v$. This result shows that the time constant is universal and depends only on the value of $I$. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of $I$, under the optimal moment condition on $F$. The proof method also shows an analogous universality on other two-dimensional lattices, assuming the time constant exists.

math.PR↗

arXiv2019

Zero-temperature Glauber dynamics on the 3-regular tree and the median process

In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d.~initial spins $σ_x(0)$ from $\{-1,+1\}$ with $\mathbb{P}_p(σ_x(0) = +1)=p$, and they update their spins at the arrival times of i.i.d. Poisson processes to agree with a majority of their neighbors. We study this process on the 3-regular tree $\mathbb{T}_3$, where it is known that the critical threshold $p_c$, below which $\mathbb{P}_p$-a.s. all spins fixate to $-1$, is strictly less than $1/2$. Defining $θ(p)$ to be the $\mathbb{P}_p$-probability that a vertex fixates to $+1$, we show that $θ$ is a continuous function on $[0,1]$, so that, in particular, $θ(p_c)=0$. To do this, we introduce a new continuous-spin process we call the median process, which gives a coupling of all the measures $\mathbb{P}_p$. Along the way, we study the time-infinity agreement clusters of the median process, show that they are a.s. finite, and deduce that all continuous spins flip finitely often. In the second half of the paper, we show a correlation decay statement for the discrete spins under $\mathbb{P}_p$ for a.e. value of $p$. The proof relies on finiteness of a vertex's "trace" in the median process to derive a stability of discrete spins under finite resampling. Last, we use our methods to answer a question of C. Howard (2001) on the emergence of spin chains in $\mathbb{T}_3$ in finite time.

math.PR↗

arXiv2019

Sublinear variance in Euclidean first-passage percolation

The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this inequality to $Cn/\log n$. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the passage time as a function of Bernoulli sequences and uniform sequences, and develop several "greedy lattice animal" arguments.

math.PR↗

arXiv2018

Parking on transitive unimodular graphs

Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p \geq 1/2$, and only finitely many times otherwise.

math.PR↗