arXiv subjects
Publications and source records attributed to Gideon Amir.
We give lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear (degree 1) and quadratic (degree 2) mother groups on the orbit of the zero ray. These bounds, combined with results of \cite{JNS} show that every quadratic activity automaton group is amenable. The resistance bounds use an apparently new "weighted" version of the Nash-Williams criterion which may be of independent interest.
We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy $\{f(n)\}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that $Cn^{d-2}$ firefighters are enough, answering and strengthening a conjecture of Develin and Hartke. We also prove that intermediate growth Cayley graphs do not satisfy polynomial containment, and give explicit lower bounds depending on the growth rate of the group. These bounds can be further improved when more geometric information is available, such as for Grigorchuk's group.
We consider two-dimensional dependent dynamical site percolation where sites perform majority dynamics. We introduce the critical percolation function at time t as the infimum density with which one needs to begin in order to obtain an infinite open component at time t. We prove that, for any fixed time t, there is no percolation at criticality and that the critical percolation function is continuous. We also prove that, for any positive time, the percolation threshold is strictly smaller than the critical probability for independent site percolation.
A combinatorial approach is used to study the critical behavior of a $q$-state Potts model with a round-the-face interaction. Using this approach it is shown that the model exhibits a first order transition for $q>3$. A second order transition is numerically detected for $q=2$. Based on these findings, it is deduced that for some two-dimensional ferromagnetic Potts models with completely local interaction, there is a changeover in the transition order at a critical integer $q_c\leq 3$. This stands in contrast to the standard two-spin interaction Potts model where the maximal integer value for which the transition is continuous is $q_c=4$. A lower bound on the first order critical temperature is additionally derived.
In [AAV] Amir, Angel and Valk{ó} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-type versions of the TAZRP; for example we are able to give a precise description of the contents of a single site in equilibrium for a multi-type TAZRP with continuous labels.
We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, out results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive group of automorphisms for which the directed balls grow exponentially.
We introduce and study the mutating contact process, a variant of the multitype contact process, where one type mutates at a constant rate to the other type. We prove that on $\mathbb{Z}$ a single mutant cannot survive while on $\mathbb{T}_{d}$ there are distinct weak survival and extinction of a single mutant phases, yet the limiting distribution concentrates on configurations with no mutants of the first type for any values of the parameters.
In this paper we introduce the notion of Random Walk in Changing Environment - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and edge sets but with different (possibly 0) weights in each step. This is a very wide class of RW, which includes some well known types of RW as special cases (e.g. reinforced RW, true SAW). We define and explore various possible properties of such walks, and provide criteria for recurrence and transience when the underlying graph is $\mathbb{N}$ or a tree. We provide an example of such a process on $\mathbb{Z}^2$ where conductances can only change from $1$ to $2$ (once for each edge) but nevertheless the walk is transient, and conjecture that such behaviour cannot happen when the weights are chosen in advance, that is, do not depend on the location of the RW.
For any order of growth $f(n)=o(\log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function with slower growth. The construction uses permutational wreath products in which the base group is defined via its properly chosen Schreier graph.
Consider several independent Poisson point processes on R^d, each with a different colour and perhaps a different intensity, and suppose we are given a set of allowed family types, each of which is a multiset of colours such as red-blue or red-red-green. We study translation-invariant schemes for partitioning the points into families of allowed types. This generalizes the 1-colour and 2-colour matching schemes studied previously (where the sets of allowed family types are the singletons {red-red} and {red-blue} respectively). We characterize when such a scheme exists, as well as the optimal tail behaviour of a typical family diameter. The latter has two different regimes that are analogous to the 1-colour and 2-colour cases, and correspond to the intensity vector lying in the interior and boundary of the existence region respectively. We also address the effect of requiring the partition to be a deterministic function (i.e. a factor) of the points. Here we find the optimal tail behaviour in dimension 1. There is a further separation into two regimes, governed by algebraic properties of the allowed family types.
For every 3/4 <= beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any nice prescribed function up to a constant factor.
For every $3/4\le δ, β< 1$ satisfying $δ\leq β< \frac{1+δ}{2}$ we construct a finitely generated group $Γ$ and a (symmetric, finitely supported) random walk $X_n$ on $Γ$ so that its expected distance from its starting point satisfies $E|X_n|\asymp n^β$ and its entropy satisfies $H(X_n)\asymp n^δ$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^γ$ for some $γ<1$.
We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure.
In this paper we study the structure of the limit aggregate $A_\infty = \bigcup_{n\geq 0} A_n$ of the one-dimensional long range diffusion limited aggregation process defined in [AABK09]. We show (under some regularity conditions) that for walks with finite third moment $A_\infty$ has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and $A_\infty$ has 0 density. We define a tree structure on the aggregates and show some results on the degrees and number of ends of these random trees. We introduce a new "harmonic competition" model where different colours compete for harmonic measure, and show how the tree structure is related to coexistence in this model.
The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner [8].
We show that for an i.i.d. bounded and weakly elliptic cookie environment, one dimensional excited random walk on the $k$-time leftover environment is right transient if and only if $δ> k+1$ and has positive speed if and only if $δ> k+2$, where $δ$ is the expected drift per site. This gives, to the best of our knowledge, the first example of an environment with positive speed that has stationary and ergodic properties but does not follow by trivial comparison to an i.i.d. environment. In another formulation, we show that on such environments an excited mob of $k$ walkers is right transient if and only if $δ> k$ and moves with positive speed if and only if $δ>k+1$. We show that for stationary and erogodic cookie environments, a law of large numbers for the walkers on leftover environments holds. Whenever the environments are also elliptic, a zero-one law for directional transience is also proven.
We examine diffusion-limited aggregation for a one-dimensional random walk with long jumps. We achieve upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. In this paper we handle the case of transient walks.
In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1,1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of Z, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.