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Edwin Perkins

Publications and source records attributed to Edwin Perkins.

15 recordsLinked to original sources
arXiv2025

Random skeletons in high-dimensional lattice trees

We study the behaviour of the rescaled minimal subtree containing the origin and K random vertices selected from a random critical (sufficiently spread-out, and in dimensions d > 8) lattice tree conditioned to survive until time ns, in the limit as n goes to infinity. We prove joint weak convergence of various quantities associated with these subtrees under this sequence of conditional measures to their counterparts for historical Brownian motion. We also show that when K is sufficiently large the entire rescaled tree is close to this rescaled skeleton with high probability, uniformly in n. These two results are the key conditions used in [5] to prove that the simple random walk on sufficiently spread-out lattice trees (conditioned to survive for a long time) converges to Brownian motion on a super-Brownian motion (conditioned to survive). The main convergence result is established more generally for a sequence of historical processes converging to historical Brownian motion in the sense of finite dimensional distributions and satisfying a pair of technical conditions. The conditions are readily verified for the lattice trees mentioned above and also for critical branching random walk. We expect that it will also apply with suitable changes to other lattice models in sufficiently high dimensions such as oriented percolation and the voter model. In addition some forms of the second skeleton density result are already established in this generality.

math.PR↗
arXiv2024

Total disconnectedness and percolation for the supports of super-tree random measures

Super-tree random measures (STRMs) were introduced by Allouba, Durrett, Hawkes and Perkins as a simple stochastic model which emulates a superprocess at a fixed time. A STRM $\nu$ arises as the a.s. limit of a sequence of empirical measures for a discrete time particle system which undergoes independent supercritical branching and independent random displacement (spatial motion) of children from their parents. We study the connectedness properties of the closed support of a STRM ($\mathrm{supp}(\nu)$) for a particular choice of random displacement. Our main results are distinct sufficient conditions for the a.s. total disconnectedness (TD) of $\mathrm{supp}(\nu)$, and for percolation on $\mathrm{supp}(\nu)$ which will imply a.s. existence of a non-trivial connected component in $\mathrm{supp}(\nu)$. We illustrate a close connection between a subclass of these STRM's and super-Brownian motion (SBM). For these particular STRM's the above results give a.s. TD of the support in three and higher dimensions and the existence of a non-trivial connected component in two dimensions, with the three-dimensional case being critical. The latter two-dimensional result assumes that $p_c(\mathbb{Z}^2)$, the critical probability for site percolation on $\mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence supporting this condition although the known rigorous bounds fall just short.) This gives evidence that the same connectedness properties should hold for SBM. The latter remains an interesting open problem in dimensions $2$ and $3$ ever since it was first posed by Don Dawson over $30$ years ago.

math.PR↗
arXiv2020

Optimal anytime regret with two experts

We consider the classical problem of prediction with expert advice. In the fixed-time setting, where the time horizon is known in advance, algorithms that achieve the optimal regret are known when there are two, three, or four experts or when the number of experts is large. Much less is known about the problem in the anytime setting, where the time horizon is not known in advance. No minimax optimal algorithm was previously known in the anytime setting, regardless of the number of experts. Even for the case of two experts, Luo and Schapire have left open the problem of determining the optimal algorithm. We design the first minimax optimal algorithm for minimizing regret in the anytime setting. We consider the case of two experts, and prove that the optimal regret is $\gamma \sqrt{t} / 2$ at all time steps $t$, where $\gamma$ is a natural constant that arose 35 years ago in studying fundamental properties of Brownian motion. The algorithm is designed by considering a continuous analogue of the regret problem, which is solved using ideas from stochastic calculus.

cs.LG↗
arXiv2018

On the topological boundary of the range of super-Brownian motion-extended version

We show that if $\partial\mathcal{R}$ is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$ implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases} 4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{ if }d=3. \end{cases}$$ This improves recent results of the last two authors (arxiv:1711.03486) by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.

math.PR↗
arXiv2018

On the range of lattice models in high dimensions - extended version

In this paper we investigate the scaling limit of the range (the set of visited vertices) for a class of critical lattice models, starting from a single initial particle at the origin. We give conditions on the random sets and an associated "ancestral relation" under which, conditional on longterm survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of exiting a large ball, that is for the extrinsic one-arm probabililty. We show that these conditions are satisfied by the voter model in dimensions $d \ge 2$ and critical sufficiently spread out lattice trees in dimensions $d > 8$. The latter result also has important consequences for the behaviour of random walks on lattice trees in high dimensions. We conjecture that our conditions are also satisfied by other models (at criticality above the critical dimension) such as sufficiently spread out oriented percolation and contact processes in dimensions d > 4. This version of the paper contains details not present in the submitted version.

math.PR↗
arXiv2018

On the boundary of the zero set of super-Brownian motion and its local time

If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2\lambda_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-\lambda_0\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins who proved the above with positive probability. To establish this result we derive some new basic properties of a recently introduced boundary local time and analyze the behaviour of $X(t,\cdot)$ near the upper edge of its support. Numerical estimates of $\lambda_0$ suggest that the above Hausdorff dimension is approximately $.224$.

math.PR↗
arXiv2016

A lower bound for $p_c$ in range-$R$ bond percolation in two and three dimensions

We use the connection between bond percolation and SIR epidemics to establish lower bounds for the critical percolation probability in $2$ and $3$ dimensions as the range becomes large. The bound agrees with the conjectured asymptotics for the long range critical probability, refines results of M. Penrose, and complements results of van der Hofstad and Sakai in dimensions greater than $6$.

math.PR↗
arXiv2015

On the boundary of the support of super-Brownian motion: with appendices

We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x)<a) as a approaches 0, as well as the Hausdorff dimension of the boundary of the support of X(t). The answers are in terms of the lead eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.

math.PR↗
arXiv2012

Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon$-H\"older diffusion coefficients

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x) +\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$ is H\"{o}lder continuous of index $\gamma>3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE's is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $\gamma=1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

math.PR↗
arXiv2011

Voter Model Perturbations and Reaction Diffusion Equations

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.

math.PR↗
arXiv2007

Weak convergence of measure-valued processes and $r$-point functions

We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the $r$-point functions and perhaps convergence of the ``survival probabilities.'' These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.

math.PR↗
arXiv2005

On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients

We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on IR^d for d greater or equal to 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hoelder continuity of the solutions near zero is established.

math.PR↗