arXiv subjects

Alan Hammond

Publications and source records attributed to Alan Hammond.

At least 37 records · Page 2Linked to original sources

arXiv2019

Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two-thirds power of the interpolating distance. This two-thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit-order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of $k$ disjoint polymers crossing a unit-order region while beginning and ending within a short distance $ε$ of each other is bounded above by $ε^{(k^2 - 1)/2 \, + \, o(1)}$. This result, which we conjecture to be sharp, yields understanding of the uniform nature of the coalescence structure of polymers, and plays a foundational role in [Ham17c] in proving comparison on unit-order scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an on-scale articulation of the two-thirds power law for polymer geometry: polymers fluctuate by $ε^{2/3}$ on short scales $ε$.

math.PR↗

arXiv2019

Modulus of continuity of polymer weight profiles in Brownian last passage percolation

In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontally, over $(z,1)$, $z \in \mathbb{R}$, so that the polymer weight profile may be studied as a function of $z \in \mathbb{R}$. This profile is known to manifest a one-half power law, having $1/2-$-Hölder continuity. The polymer weight profile may be defined beginning from a much more general initial condition. In this article, we present a more general assertion of this one-half power law, as well as a bound on the poly-logarithmic correction. For a very broad class of initial data, the polymer weight profile has a modulus of continuity of the order of $x^{1/2} \big( \log x^{-1} \big)^{2/3}$, with a high degree of uniformity in the scaling parameter and the initial condition.

math.PR↗

arXiv2019

Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE

The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey's principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.

math.PR↗

arXiv2018

Self-attracting self-avoiding walk

This article is concerned with self-avoiding walks (SAW) on $\mathbb{Z}^{d}$ that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions. This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in $d\geq 5$, carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander.

math.PR↗

arXiv2018

On self-avoiding polygons and walks: the snake method via polygon joining

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z}^d$, and write $Γ$ for a $\mathsf{W}_n$-distributed walk. We show that the closing probability $\mathsf{W}_n \big( \vert \vert Γ_n \vert \vert = 1 \big)$ that $Γ$'s endpoint neighbours the origin is at most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d = 2$. This result is proved using the snake method, a technique for proving closing probability upper bounds, which originated in [3] and was made explicit in [8]. Our conclusion is reached by applying the snake method in unison with a polygon joining technique whose use was initiated by Madras in [13].

math.PR↗

arXiv2018

On self-avoiding polygons and walks: the snake method via pattern fluctuation

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\mathbb{Z}^d$, and write $Γ$ for a $\mathsf{W}_n$-distributed walk. We show that the closing probability $\mathsf{W}_n \big( \vert\vert Γ_n \vert\vert = 1 \big)$ that $Γ$'s endpoint neighbours the origin is at most $n^{-1/2 + o(1)}$ in any dimension $d \geq 2$. The method of proof is a reworking of that in [4], which found a closing probability upper bound of $n^{-1/4 + o(1)}$. A key element of the proof is made explicit and called the snake method. It is applied to prove the $n^{-1/2 + o(1)}$ upper bound by means a technique of Gaussian pattern fluctuation.

math.PR↗

arXiv2018

An upper bound on the number of self-avoiding polygons via joining

For $d \geq 2$ and $n \in \mathbb{N}$ even, let $p_n = p_n(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^d$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n \in 2\mathbb{N}} p_n^{1/n} \in (0,\infty)$ is called the connective constant and denoted by $μ$. Madras [J. Statist. Phys. 78 (1995) no. 3--4, 681--699] has shown that $p_n μ^{-n} \leq C n^{-1/2}$ in dimension $d=2$. Here we establish that $p_n μ^{-n} \leq n^{-3/2 + o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of self-avoiding walk and argue that, when $d \geq 3$, an upper bound of $n^{-2 + d^{-1} + o(1)}$ holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.

math.PR↗

arXiv2018

Critical point for infinite cycles in a random loop model on trees

We study a spatial model of random permutations on trees with a time parameter $T>0$, a special case of which is the random stirring process. The model on trees was first analysed by Björnberg and Ueltschi[BU16], who established the existence of infinite cycles for $T$ slightly above a putatively identified critical value but left open behaviour at arbitrarily high values of $T$. We show the existence of infinite cycles for all $T$ greater than a constant, thus classifying behaviour for all values of $T$ and establishing the existence of a sharp phase transition. Numerical studies [BBBU15] of the model on $\mathbb{Z}^d$ have shown behaviour with strong similarities to what is proven for trees.

math.PR↗

arXiv2018

Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation $t$ (and a horizontal separation of the same order), the maximum transversal fluctuation has order $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$, $z\in [1,2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3}\big(\log t^{-1}\big)^{2/3}$. In this way, we identify exponent pairs of $(2/3,1/3)$ and $(1/3,2/3)$ in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [10,11,9] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.

math.PR↗

arXiv2017

The Competition of Roughness and Curvature in Area-Constrained Polymer Models

The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarised by an exponent triple $(1/2,1/3,2/3)$ representing local interface fluctuation, local roughness (or inward deviation) and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models (Alexander, '01, Hammond, '11,'12). In this article, we offer a new perspective on this phenomenon. We consider directed last passage percolation model in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature as before, but now its local structure is the KPZ fixed point polymer's rather than Brownian. The local interface fluctuation exponent is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path's convex hull are governed by an exponent of $3/4$, and inward deviation by an exponent of $1/2$. That is, the exponent triple is now $(2/3,1/2,3/4)$ in place of $(1/2,1/3,2/3)$. This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established (Biskup, Louidor, Procaccia and Rosenthal, '12).

math.PR↗

arXiv2017

On self-avoiding polygons and walks: counting, joining and closing

For d at least two and integer n, let c_n = c_n(d) denote the number of length n self-avoiding walks beginning at the origin in the integer lattice Z^d, and, for even n, let p_n = p_n(d) denote the number of length n self-avoiding polygons in Z^d up to translation. Then the probability under the uniform law W_n on self-avoiding walks Gamma of any given odd length n beginning at the origin that Gamma closes -- i.e., that Gamma's endpoint is a neighbour of the origin -- is given by W_n ( Gamma closes ) = 2(n+1) p_{n+1}/c_n. The polygon and walk cardinalities share a common exponential growth: lim_n c_n^{1/n} = lim_{n even} p_n^{1/n} = mu (where the common value mu is called the connective constant). Madras [26] has shown that p_n is at most C n^{-1/2} mu^n in dimension d=2, while the closing probability was recently shown in [12] to satisfy W_n ( Gamma closes ) is at most n^{-1/4 + o(1)} in any dimension d at least two. Here we establish that (1) W_n ( Gamma closes ) is at most n^{-1/2 + o(1)} for any d at least two; (2) W_n ( Gamma closes ) is at most n^{-4/7 + o(1)} for a subsequence of odd n, if d = 2; and (3) p_n is at most n^{-3/2 + o(1)} mu^n for a set of even n of full density when d=2. We also argue that the closing probability is bounded above by n^{-(1 - 1/d) + o(1)} on a full density set when d is at least three for a certain variant of self-avoiding walk.

math.PR↗

arXiv2014

On the probability that self-avoiding walk ends at a given point

We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 + epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the probability that self-avoiding walk is a polygon.

math.PR↗

arXiv2013

Sharp phase transition in the random stirring model on trees

We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d_0 such that, for any d \geq d_0, the set of parameter values at which the random stirring model on the rooted regular tree with offspring degree d almost surely contains an infinite cycle consists of a semi-infinite interval. The critical point at the left-hand end of this interval is at least 1/d + 1/(2d^2) and at most 1/d + 2/(d^2). This version is a major revision, with a much shorter proof. Principal among the changes are a reworking of the argument in Section 4 of the old version, which was proposed by a referee, and the use of a simpler means of handling a boundary case, which eliminates the previous Section 6.

math.PR↗

arXiv2013

Phase transition for the speed of the biased random walk on the supercritical percolation cluster

We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one-dimensional; we prove such a `dynamic renormalization' statement in a much stronger form than was previously known.

math.PR↗

arXiv2013

Infinite cycles in the random stirring model on trees

We prove that, in the random stirring model of parameter T on an infinite rooted tree each of whose vertices has at least two offspring, infinite cycles exist almost surely, provided that T is sufficiently high. In the appendices, the bound on degree above which the result holds is improved slightly.

math.PR↗

arXiv2013

The kinetic limit of a system of coagulating planar Brownian particles

We study a model of mass-bearing coagulating planar Brownian particles. Coagulation is prone to occur when two particles become within a distance of order $ε$. We assume that the initial number of particles is of the order of $| \log ε|. Under suitable assumptions on the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.

math.PR↗

arXiv2013

Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster

In dynamical critical site percolation on the triangular lattice or bond percolation on \Z^2, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten's Incipient Infinite Cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We also study the collapse of the infinite cluster near typical exceptional times, and establish a relation between static and dynamic exponents, analogous to Kesten's near-critical relation.

math.PR↗

arXiv2013

Brownian Gibbs property for Airy line ensembles

Consider N Brownian bridges B_i:[-N,N] -> R, B_i(-N) = B_i(N) = 0, 1 <= i <= N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a limit as N -> infinity of these curves scaled around (0,2^{1/2} N) horizontally by a factor of N^{2/3} and vertically by N^{1/3}. If a parabola is added to each limit curve, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which curves are a.s. everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with "wanderers" and "outliers". We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property -- called the Brownian Gibbs property -- of being invariant under the following action. Select an index 1 <= k <= N and erase B_k on a fixed time interval (a,b) subset of (-N,N); then replace this erased curve with a new curve on (a,b) according to the law of a Brownian bridge between the two existing endpoints (a,B_k(a)) and (b,B_k(b)), conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edge-scaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence is a proof of M. Prahofer and H. Spohn's prediction that the lines of the Airy line ensemble are locally absolutely continuous with respect to Brownian motion. We also prove the conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point, thus establishing the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights.

math.PR↗