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Kenneth S. Alexander

Publications and source records attributed to Kenneth S. Alexander.

At least 19 recordsLinked to original sources
arXiv2023

Properties of first passage percolation above the (hypothetical) critical dimension

It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\mathbb{Z}^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $χ=0$ in dimensions $d>d_c$. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from $d 0$ must be false if $χ=0$. A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false. These consequences support the idea that when $χ=0$, passage times are ``local'' in the sense that the passage time from $x$ to $y$ is primarily determined by the configuration near $x$ and $y$. Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.

math.PR↗
arXiv2022

Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension

We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic properties conjectured to be true, and derive consequences from these properties. The assumptions are roughly as follows: (i) the fluctuation scale $σ(r)$ of the passage time on scale $r$ grows approximately as a positive power $r^χ$, in the sense that two natural definitions of $σ(r)$ and $χ$ yield the same value $χ$, and (ii) the limit shape boundary has curvature uniformly bounded away from 0 and $\infty$ (a requirement we can sometimes limit to a neighborhood of some fixed direction.) The main a.s. consequences derived are the following, with $ν$ denoting a subpolynomial function and $ξ=(1+χ)/2$ the transverse wandering exponent: (a) for one-ended geodesic rays with a given asymptotic direction $θ$, starting in a natural halfspace $H$, for the hyperplane at distance $r$ from $H$, the density of "entry points" where some geodesic ray first crosses the hyperplane is at most $ν(r)/r^{(d-1)ξ}$, (b) the system has no bigeodesics, i.e. two-ended infinite geodesics, (c) given two sites $x,y$, and a third site $z$ at distance at least $\ell$ from $x$ and $y$, the probability that the geodesic from $x$ to $y$ passes through $z$ is at most $ν(\ell)/\ell^{(d-1)ξ}$, and (d) in $d=2$, the probability that the geodesic rays in a given direction from two sites have not coalesced after distance $r$ decays like $r^{-ξ}$ to within a subpolynomial factor. Our entry-point density bound compares to a natural conjecture of $c/r^{(d-1)ξ}$, corresponding to a spacing of order $r^ξ$ between entry points, which is the conjectured scale of the transverse wandering.

math.PR↗
arXiv2021

Uniform fluctuation and wandering bounds in first passage percolation

We consider first passage percolation on certain isotropic random graphs in $\mathbb{R}^d$. We assume exponential concentration of passage times $T(x,y)$, on some scale $σ_r$ whenever $|y-x|$ is of order $r$, with $σ_r$ "growning like $r^χ$" for some $0<χ<1$. Heuristically this means transverse wandering of geodesics should be at most of order $Δ_r = (rσ_r)^{1/2}$. We show that in fact uniform versions of exponential concentration and wandering bounds hold: except with probability exponentially small in $t$, there are no $x,y$ in a natural cylinder of length $r$ and radius $KΔ_r$ for which either (i) $|T(x,y) - ET(x,y)|\geq tσ_r$, or (ii) the geodesic from $x$ to $y$ wanders more than distance $\sqrt{t}Δ_r$ from the cylinder axis. We also establish that for the time constant $μ= \lim_n ET(0,ne_1)/n$, the "nonrandom error" $|μ|x| - ET(0,x)|$ is at most a constant multiple of $σ(|x|)$.

math.PR↗
arXiv2018

Geodesics Toward Corners in First Passage Percolation

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg and Hoffman (preprint, 2016), and others. However the main results do not cover geodesics in the direction of corners of the limit shape $\mathcal{B}$, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics. Consequently, the set of points of $\partial \mathcal{B}$ which are in the direction of some geodesic does not have all of $\mathcal{B}$ as its convex hull.

math.PR↗
arXiv2017

Pinning of a renewal on a quenched renewal

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $σ$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $σ$ have infinite mean. The "polymer" -- of length $σ_N$ -- is given by another renewal $τ$, whose law is modified by the Boltzmann weight $\exp(β\sum_{n=1}^N \mathbf{1}_{\{σ_n\inτ\}})$. Our assumption is that $τ$ and $σ$ have gap distributions with power-law-decay exponents $1+α$ and $1+\tilde α$ respectively, with $α\geq 0,\tilde α>0$. There is a localization phase transition: above a critical value $β_c$ the free energy is positive, meaning that $τ$ is \emph{pinned} on the quenched renewal $σ$. We consider the question of relevance of the disorder, that is to know when $β_c$ differs from its annealed counterpart $β_c^{\rm ann}$. We show that $β_c=β_c^{\rm ann}$ whenever $ α+\tilde α\geq 1$, and $β_c=0$ if and only if the renewal $τ\capσ$ is recurrent. On the other hand, we show $β_c>β_c^{\rm ann}$ when $ α+\frac32\, \tilde α<1$. We give evidence that this should in fact be true whenever $ α+\tilde α<1$, providing examples for all such $ α,\tilde α$ of distributions of $τ,σ$ for which $β_c>β_c^{\rm ann}$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($σ_N=τ_N$), and one in which the polymer length is $τ_N$ rather than $σ_N$. In both cases we show the critical point is the same as in the original model, at least when $ α>0$.

math.PR↗
arXiv2016

Local limit theorems and renewal theory with no moments

We study i.i.d. sums $τ_k$ of nonnegative variables with index $0$: this means $\mathbf{P}(τ_1=n) = φ(n) n^{-1}$, with $φ(\cdot)$ slowly varying, so that $\mathbf{E}(τ_1^\varepsilon)=\infty$ for all $\varepsilon>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of $\mathbf{P}(τ_k=n)$ when $n$ is at least the typical length of $τ_k$. A recent renewal theorem by Nagaev [21] is an immediate consequence: $\mathbf{P}(n\inτ) \sim \mathbf{P}(τ_1=n)/\mathbf{P}(τ_1 > n)^2$ as $n\to\infty$. If instead we only assume regular variation of $\mathbf{P}(n\inτ)$ and slow variation of $U_n:= \sum_{k=0}^n \mathbf{P}(k\inτ)$, we obtain a similar equivalence but with $\mathbf{P}(τ_1=n)$ replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of $\mathbf{P}(τ_k \leq n)$ when $n$ is much smaller than the typical length of $τ_k$.

math.PR↗
arXiv2016

Local asymptotics for the first intersection of two independent renewals

We study the intersection of two independent renewal processes, $ρ=τ\capσ$. Assuming that $\mathbf{P}(τ_1 = n ) = φ(n)\, n^{-(1+α)}$ and $\mathbf{P}(σ_1 = n ) = \tildeφ(n)\, n^{-(1+ \tildeα)} $ for some $α,\tilde α\geq 0$ and some slowly varying $φ,\tildeφ$, we give the asymptotic behavior first of $\mathbf{P}(ρ_1>n)$ (which is straightforward except in the case of $\min(α,\tildeα)=1$) and then of $\mathbf{P}(ρ_1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities $\mathbf{P}(ρ_1=n)$ while knowing asymptotically the renewal mass function $\mathbf{P}(n\inρ)=\mathbf{P}(n\inτ)\mathbf{P}(n\inσ)$. Our results can be used to bound coupling-related quantities, specifically the increments $|\mathbf{P}(n\inτ)-\mathbf{P}(n-1\inτ)|$ of the renewal mass function.

math.PR↗
arXiv2015

Directed polymers in a random environment with a defect line

We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $β$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $β$) of the annealed critical point $u_c^a=0$. For the case $u=0$, we show that the difference between quenched and annealed free energies is of order $β^4$ as $β\to 0$, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions.

math.PR↗
arXiv2014

Path properties of the disordered pinning model in the delocalized regime

We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense "tight in probability" as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

math.PR↗
arXiv2013

Subgaussian concentration and rates of convergence in directed polymers

We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder. We study the partition function $Z_{N,ω}$ and establish exponential concentration of $\log Z_{N,ω}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,ω}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log N}}\log \log N)$.

math.PR↗
arXiv2013

Controlled random walk with a target site

We consider a simple random walk W_i in 1 or 2 dimensions, in which the walker may choose to stand still for a limited time. The time horizon is n, the maximum consecutive time steps which can be spent standing still is m_n and the goal is to maximize P(W_n=0). We show that for dimension 1, if m_n grows faster than (\log n)^{2+γ} for some γ>0, there is a strategy for each n such that P(W_n = 0) approaches 1. For dimension 2, if m_n grows faster than a positive power of n then there are strategies keeping P(W_n=0) bounded away from 0.

math.PR↗
arXiv2010

Excursions and local limit theorems for Bessel-like random walks

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -δ/2x + o(1/x) with δ> -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}ϕ(n), where c = (3+δ)/2 and ϕis a slowly varying function given explicitly in terms of the o(1/x) terms.

math.PR↗
arXiv2010

Layering and wetting transitions for an SOS interface

We study the solid-on-solid interface model above a horizontal wall in three dimensional space, with an attractive interaction when the interface is in contact with the wall, at low temperatures. There is no bulk external field. The system presents a sequence of layering transitions, whose levels increase with the temperature, before reaching the wetting transition.

math-ph↗
arXiv2010

Equality of critical points for polymer depinning transitions with loop exponent one

We consider a polymer with configuration modelled by the trajectory of a Markov chain, interacting with a potential of form $u+V_n$ when it visits a particular state 0 at time $n$, with $\{V_n\}$ representing i.i.d. quenched disorder. There is a critical value of $u$ above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length $n$ takes the form $ϕ(n)/n$ for some slowly varying $ϕ$; this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of $u$ in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ, at least at low temperatures.

math.PR↗
arXiv2008

Quenched and Annealed Critical Points in Polymer Pinning Models

We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c}ϕ(n)$ with $c \geq 1$ and $ϕ$ slowly varying. Comparing to the corresponding annealed system, in which the $V_n$ are effectively replaced by a constant, it is known that the quenched and annealed critical points differ at all temperatures for $3/2 2$, but only at low temperatures for $c<3/2$. For high temperatures and $3/2 3/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$.

math.PR↗
arXiv2006

Ivy on the ceiling: first-order polymer depinning transitions with quenched disorder

We consider a polymer, with monomer locations modeled by the trajectory of an underlying Markov chain, in the presence of a potential thatinteracts with the polymer when it visits a particular site 0. Disorder is introduced by having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. When the excursions of the underlying chain have a finite mean but no finite exponential moment, it is known that the depinning transition (more precisely, the contact fraction) in the corresponding annealed system is discontinuous. One generally expects the presence of disorder to smooth transitions, and it was proved by Giacomin and Toninelli that when the excursion length distribution has power-law tails, the quenched system has a continuous transition even if the annealed system does not. We show here that when the underlying chain is transient but the finite part of the excursion length distribution has exponential tails, then the depinning transition is discontinuous even in the quenched system, and the quenched and annealed critical points are strictly different. By contrast, in the recurrent case, the depinning behavior depends on the subexponential prefactors on the exponential decay of the excursion length distribution, and when these prefactors decay with an appropriate power law, the quenched transition is continuous even though the annealed one is not.

math.PR↗
arXiv2006

The Effect of Disorder on Polymer Depinning Transitions

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length $n$ is given by $n^{-c}ϕ(n)$ for some $1 3/2$, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point--the size of this neighborhood scales as $β^{1/(2c-3)}$ where $β$ is the inverse temperature. For $c<3/2$, given $ε>0$, for sufficiently high temperature the quenched and annealed curves are within a factor of $1-ε$ for all $u$ near the critical point; in particular the quenched and annealed critical points are equal. For $c=3/2$ the regime depends on the slowly varying function $ϕ$.

math.PR↗