arXiv subjects

Jack Hanson

Publications and source records attributed to Jack Hanson.

30 records · Page 2Linked to original sources

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

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

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 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↗

arXiv2014

Rate of convergence of the mean for sub-additive ergodic sequences

For 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 $γ$ given roughly by $\mathbb{E}X_{0,n} \sim ng + n^γ$, and, assuming existence of a fluctuation exponent $χ$ that gives $\mathrm{Var}~X_{0,n} \sim n^{2χ}$, we provide a lower bound for $γ$ of the form $γ\geq χ$. The main requirement is that $χ\neq 1/2$. In the case $χ=1/2$ and under the assumption $\mathrm{Var}~X_{0,n} = O(n/(\log n)^β)$ for some $β>0$, we prove $γ\geq χ- c(β)$ for a $β$-dependent constant $c(β)$. These results show in particular that non-diffusive fluctuations are associated to non-trivial $γ$. 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 $χ$ exists. In the case of first-passage percolation in $\mathbb Z^d$, we provide a version of $γ\geq -1/2$ without assuming existence of $χ$.

math.PR↗

arXiv2012

Subdiffusivity of random walk on the 2D invasion percolation cluster

We derive "quenched" subdiffusive lower bounds for the exit time tau(n) from a box of size n for the simple random walk on the planar invasion percolation cluster. The first part of the paper is devoted to proving an almost sure analog of H. Kesten's subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on the instrinsic distance in these graphs and general inequalities for reversible Markov chains. In the second part of the paper, we present a sharpening of Kesten's original argument, leading to an explicit almost sure lower bound for tau(n) in terms of percolation arm exponents. The methods give tau(n) \geq n^{2+epsilon_0 + kappa}, where epsilon_0>0 depends on the instrinsic distance and (assuming the exact value of the backbone exponent) kappa can be taken to be 17/384 on the hexagonal lattice.

math.PR↗

arXiv2012

Busemann functions and infinite geodesics in two-dimensional first-passage percolation

We study first-passage percolation on Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of infinite geodesics. Some of these were studied in the late 90's by C. Newman and collaborators under strong assumptions on the limiting shape and weight distribution. In this paper we develop a framework for working with distributional limits of Busemann functions and use it to prove forms of Newman's results under minimal assumptions. For instance, we show a form of coalescence of long finite geodesics in any deterministic direction. We also introduce a purely directional condition which replaces Newman's global curvature condition and whose assumption we show implies the existence of directional geodesics. Without this condition, we prove existence of infinite geodesics which are directed in sectors. Last, we analyze distributional limits of geodesic graphs, proving almost-sure coalescence and nonexistence of infinite backward paths. This result relates to the conjecture of nonexistence of "bigeodesics."

math.PR↗