Search ResultsShowing 1-20 of 25
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arXiv:2308.10114 (Published 2023-08-19)
Exceptional behavior in critical first-passage percolation and random sums
Comments: 28 pages, 2 figuresCategories: math.PRWe 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.
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arXiv:2208.11576 (Published 2022-08-24)
The number of geodesics in planar first-passage percolation grows sublinearly
Comments: 49 pages, 3 figures. A video summary may be found at: https://youtu.be/vzrO1iKIen4Categories: math.PRWe study a random perturbation of the Euclidean plane, and show that it is unlikely that the distance-minimizing path between the two points can be extended into an infinite distance-minimizing path. More precisely, we study a large class of planar first-passage percolation models and show that the probability that a given site is visited by an infinite geodesic starting at the origin tends to zero uniformly with the distance. In particular, this show that the collection of infinite geodesics starting at the origin covers a negligible fraction of the plane. This provides the first progress on the `highways and byways' problem, posed by Hammersley and Welsh in the 1960s.
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arXiv:2108.13248 (Published 2021-08-30)
Transitions for exceptional times in dynamical first-passage percolation
Comments: 38 pages, 3 figuresIn first-passage percolation (FPP), we let $(\tau_v)$ 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 $\tau_v$, 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.
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arXiv:2107.14347 (Published 2021-07-29)
Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation
Comments: 63 pages, 6 figuresIn high dimensional percolation at parameter $p < p_c$, the one-arm probability $\pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $\pi_p(n) / \pi_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter $n$ box on scale $n^{d-6}$; this result complements a lower bound of Aizenman.
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arXiv:2006.16347 (Published 2020-06-29)
Random nearest neighbor graphs: the translation invariant case
Comments: 33 pages, 3 figuresCategories: math.PRIf $(\omega(e))$ 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 $\omega(\{x,y\})$ 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.
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arXiv:2003.03367 (Published 2020-03-06)
Absence of backward infinite paths for first-passage percolation in arbitrary dimension
Comments: 29 pages, 4 figuresSubjects: 60K35In 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.
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arXiv:1909.09860 (Published 2019-09-21)
Absence of Disorder Chaos for Ising Spin Glasses on $\mathbb Z^d$
We identify simple mechanisms that prevents the onset of disorder chaos for the Ising spin glass model on $\mathbb Z^d$. This was first shown by Chatterjee in the case of Gaussian couplings. We present three proofs of the theorem for general couplings with continuous distribution based on the presence in the coupling realization of stabilizing features of positive density.
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arXiv:1904.12009 (Published 2019-04-26)
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 \[ \lim_{n \to \infty} \frac{T(0,\partial B(n))}{\log n} = \frac{I}{2\sqrt{3}\pi} \text{ almost surely}, \] 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.
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arXiv:1810.08270 (Published 2018-10-18)
Lower bounds for fluctuations in first-passage percolation for general distributions
Comments: 27 pagesCategories: math.PRIn 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).
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arXiv:1810.03750 (Published 2018-10-08)
Restricted percolation critical exponents in high dimensions
Comments: 38 pages, 4 figuresDespite great progress in the study of critical percolation on $\mathbb{Z}^d$ for $d$ large, properties of critical clusters in high-dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high-dimensional critical exponents; see in particular the conjecture by Kozma-Nachmias that the probability that $0$ and $(n, n, n, \ldots)$ are connected within $[-n,n]^d$ scales as $n^{-2-2d}$. In this paper, we study the properties of critical clusters in high-dimensional half-spaces and boxes. In half-spaces, we show that the probability of an open connection ("arm") from $0$ to the boundary of a sidelength $n$ box scales as $n^{-3}$. We also find the scaling of the half-space two-point function (the probability of an open connection between two vertices) and the tail of the cluster size distribution. In boxes, we obtain the scaling of the two-point function between vertices which are any macroscopic distance away from the boundary.
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arXiv:1804.05720 (Published 2018-04-16)
Infinite geodesics, asymptotic directions, and Busemann functions in first-passage percolation
Comments: This is a chapter in a forthcoming AMS Proceedings collection of expanded notes from the AMS Short Course "Random Growth Models," which took place in Atlanta, GA, at the AMS Joint Mathematics Meetings in January 2017. Editors: Michael Damron, Firas Rassoul-Agha, Timo SeppalainenCategories: math.PRWe show existence, uniqueness, and directedness properties for infinite geodesics in the FPP model. After giving the fundamental definitions, we describe results by Newman and collaborators giving existence and uniqueness of directed geodesics under an unproven curvature assumption. We then give two proofs of the existence of at least two infinite geodesics under no unproven assumptions. In the final two sections, we give proofs of directedness statements for infinite geodesics using more recent methods which give information even under no unproven assumptions and prove a generalized uniqueness statement for infinite geodesics.
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arXiv:1709.09613 (Published 2017-09-27)
The size of the boundary in first-passage percolation
Categories: math.PRFirst-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 $t^{d-1+\alpha}$ for some $\alpha>0$ 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 $t.$
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arXiv:1708.03643 (Published 2017-08-11)
Strict inequality for the chemical distance exponent in two-dimensional critical percolation
Comments: 45 pages, 7 figuresCategories: math.PRWe 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 $Cn^{2-\delta}\pi_3(n)$, for some $\delta>0$, where $\pi_3(n)$ 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.
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arXiv:1609.06626 (Published 2016-09-21)
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.
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arXiv:1602.06475 (Published 2016-02-20)
Inequalities for critical exponents in $d$-dimensional sandpiles
Comments: 58 pagesConsider the Abelian sandpile measure on $\mathbb{Z}^d$, $d \ge 2$, obtained as the $L \to \infty$ limit of the stationary distribution of the sandpile on $[-L,L]^d \cap \mathbb{Z}^d$. When adding a grain of sand at the origin, some region topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In $d = 2$ we show that for any $1 \le k < \infty$, the last $k$ waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the radius.
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arXiv:1601.03464 (Published 2016-01-14)
On the chemical distance in critical percolation II
Comments: 24 pages, 5 figuresCategories: math.PRWe 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.
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arXiv:1512.00804 (Published 2015-12-02)
Bigeodesics in first-passage percolation
Comments: 27 pages, 6 figuresIn 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.
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arXiv:1511.03262 (Published 2015-11-10)
50 years of first passage percolation
Comments: 148 pages, 17 figuresWe 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.
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arXiv:1506.03461 (Published 2015-06-10)
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.
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arXiv:1408.4444 (Published 2014-08-19)
Rate of convergence of the mean for sub-additive ergodic sequences
Comments: 38 pages, 2 figuresCategories: math.PRFor sub-additive ergodic processes $\{X_{m,n}\}$ with weak dependence, we analyze the rate of convergence of $\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $\gamma$ given roughly by $\mathbb{E}X_{0,n} \sim ng + n^\gamma$, and, assuming existence of a fluctuation exponent $\chi$ that gives $ X_{0,n} \sim n^{2\chi}$, we provide a lower bound for $\gamma$ of the form $\gamma \geq \chi$. The main requirement is that $\chi \neq 1/2$. In the case $\chi=1/2$ and under the assumption $\mathop{\mathrm{Var}} X_{0,n} = O(n/(\log n)^\beta)$ for some $\beta>0$, we prove $\gamma \geq \chi - c(\beta)$ for a $\beta$-dependent constant $c(\beta)$. These results show in particular that non-diffusive fluctuations are associated to non-trivial $\gamma$. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming $\chi$ exists.