arXiv subjects
Publications and source records attributed to Gideon Amir.
We prove that every linear-activity automaton group is amenable. The proof is based on showing that a sufficiently symmetric random walk on a specially constructed degree 1 automaton group -- the mother group -- has asymptotic entropy 0. Our result answers an open question by Nekrashevich in the Kourovka notebook, and gives a partial answer to a question of Sidki.
Mother groups are the basic building blocks for polynomial automaton groups. We show that, in contrast with mother groups of degree 0 or 1, any bounded, symmetric, generating random walk on the mother groups of degree at least 3 has positive speed. The proof is based on an analysis of resistance in fractal mother graphs. We give upper bounds on resistances in these graphs, and show that infinite versions are tran- sient.
We consider the solution of the stochastic heat equation \partial_T \mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with delta function initial condition \mathcal{Z} (T=0)= δ_0 whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions -- the {\it crossover distributions} -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.
We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much more general processes on unimodular graphs. Roughly put, our result apply whenever the path of each particle has an automorphism invariant distribution and is independent of other particles' paths, and the interaction between particles is automorphism invariant and local. This allows us to answer a question of Rolla and Sidoravicius, in a more general setting then had been previously known (by Shellef).
We examine diffusion-limited aggregation generated by a random walk on Z with long jumps. We derive upper and lower bounds on the growth rate of the aggregate as a function of the number moments a single step of the walk has. Under various regularity conditions on the tail of the step distribution, we prove that the diameter grows as n^{beta+o(1)}, with an explicitly given beta. The growth rate of the aggregate is shown to have three phase transitions, when the walk steps have finite third moment, finite variance, and, conjecturally, finite half moment.
Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k). The same also holds when the generating set is taken to be a symmetric set of size 2k.
A random graph process, $\Gorg[1](n)$, is a sequence of graphs on $n$ vertices which begins with the edgeless graph, and where at each step a single edge is added according to a uniform distribution on the missing edges. It is well known that in such a process a giant component (of linear size) typically emerges after $(1+o(1))\frac{n}{2}$ edges (a phenomenon known as ``the double jump''), i.e., at time $t=1$ when using a timescale of $n/2$ edges in each step. We consider a generalization of this process, $\Gorg[K](n)$, which gives a weight of size 1 to missing edges between pairs of isolated vertices, and a weight of size $K \in [0,\infty)$ otherwise. This corresponds to a case where links are added between $n$ initially isolated settlements, where the probability of a new link in each step is biased according to whether or not its two endpoint settlements are still isolated. Combining methods of \cite{SpencerWormald} with analytical techniques, we describe the typical emerging time of a giant component in this process, $t_c(K)$, as the singularity point of a solution to a set of differential equations. We proceed to analyze these differential equations and obtain properties of $\Gorg$, and in particular, we show that $t_c(K)$ strictly decreases from 3/2 to 0 as $K$ increases from 0 to $\infty$, and that $t_c(K) = \frac{4}{\sqrt{3K}}(1 + o(1))$. Numerical approximations of the differential equations agree both with computer simulations of the process $\Gorg(n)$ and with the analytical results.
We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of square-root log n.
Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d > 4 there almost surely do not exist such t. Hoffman showed that for d=2 there almost surely exists t such that the random walk at time t visits the origin only finitely many times. We refine the results of Hoffman for dynamical random walk on Z^2, showing that with probability one there are times when the origin is visited only a finite number of times while other points are visited infinitely often.
In [Amir et al.], the authors consider the generalization $\Gor$ of the Erdős-Rényi random graph process $G$, where instead of adding new edges uniformly, $\Gor$ gives a weight of size 1 to missing edges between pairs of isolated vertices, and a weight of size $K\in[0,\infty)$ otherwise. This can correspond to the linking of settlements or the spreading of an epidemic. The authors investigate $\tgor(K)$, the critical time for the appearance of a giant component as a function of $K$, and prove that $\tgor=(1+o(1))\frac{4}{\sqrt{3K}}$, using a proper timescale. In this work, we show that a natural variation of the model $\Gor$ has interesting properties. Define the process $\Gand$, where a weight of size $K$ is assigned to edges between pairs of non-isolated vertices, and a weight of size 1 otherwise. We prove that the asymptotical behavior of the giant component threshold is essentially the same for $\Gand$, and namely $\tgand / \tgor$ tends to $\frac{64\sqrt{6}}{π(24+π^2)}\approx 1.47$ as $K\to\infty$. However, the corresponding thresholds for connectivity satisfy $\tcand / \tcor=\max\{{1/2},K\}$ for every $K>0$. Following the methods of [Amir et al.], $\tgand$ is characterized as the singularity point to a system of differential equations, and computer simulations of both models agree with the analytical results as well as with the asymptotic analysis. In the process, we answer the following question: when does a giant component emerge in a graph process where edges are chosen uniformly out of all edges incident to isolated vertices, while such exist, and otherwise uniformly? This corresponds to the value of $\tgand(0)$, which we show to be ${3/2}+\frac{4}{3\mathrm{e}^2-1}$.