اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدInfection spread for the frog model on trees
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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.
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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.
We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nes
ted sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.
We study the recurrence of one-per-site frog model $text{FM}(d, p)$ on a $d$-ary tree with drift parameter $pin [0,1]$, which determines the bias of frogs random walks. We are interested in the minimal drift $p_{d}$ so that the frog model is recurren
t. Using a coupling argument together with a generating function technique, we prove that for all $d ge 2$, $p_{d}le 1/3$, which is the optimal universal upper bound.
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.
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.
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