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Register a new userThe asymptotic shape theorem for the frog model on finitely generated abelian groups
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We study the frog model on Cayley graphs of groups with polynomial growth rate $D geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.
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In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices $mathds{Z}^n$ that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.
We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix $r>0$ and place a particle at each point $x$ of a unit intensity Poisson point process $mathcal P subseteq mathbb R^d - mathbb B(0,r)$. Around each point in $mathcal{P}$, put a ball of radius $r$. A particle at the origin performs Brownian motion. When it hits the ball around $x$ for some $x in mathcal P$, new particles begin independent Brownian motions from the centers of the balls in the cluster containing $x$. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For $r$ smaller than the critical threshold of continuum percolation, we show that the set of activated points in $mathcal P$ approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process.
We study the asymptotic behaviour of random walks on topological abelian groups $G$. Our main result is a sufficient condition for one random walk to overtake another in the stochastic order induced by any suitably large positive cone $G_+ subseteq G$, assuming that both walks have Radon distributions and compactly supported steps. We explain in which sense our sufficient condition is very close to a necessary one. Our result is a direct application of a recently proven theorem of real algebra, namely a Positivstellensatz for preordered semirings. It is due to Aubrun and Nechita in the one-dimensional case, but new already for $R^n$ with $n > 1$. We use our result to derive a formula for the rate at which the probabilities of a random walk decay relative to those of another, again for walks on $G$ with compactly supported Radon steps. In the case where one walk is a constant, this formula specializes to a version of Cramers large deviation theorem.
The frog model is an infection process in which dormant particles begin moving and infecting others once they become infected. We show that on the rooted $d$-ary tree with particle density $Omega(d^2)$, the set of visited sites contains a linearly expanding ball and the number of visits to the root grows linearly with high probability.
The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $mu$ on the full $d$-ary tree of height $n$. If $mu= Omega( d^2)$, all of the vertices are visited in time $Theta(nlog n)$ with high probability. Conversely, if $mu = O(d)$ the cover time is $exp(Theta(sqrt n))$ with high probability.
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