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We study the probability measure $mu_{0}$ for which the moment sequence is $binom{3n}{n}frac{1}{n+1}$. We prove that $mu_{0}$ is absolutely continuous, find the density function and prove that $mu_{0}$ is infinitely divisible with respect to the additive free convolution.
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Let $mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.
We consider the discrete-time threshold-$theta ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $theta ge 2$ and $r ge theta+2$, $epsilon_1$ is small and p is at least $p_1(epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2epsilon_1$ up to time $exp(gamma_1(r)n)$ with probability at least $1 - exp(-gamma_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $epsilon_2(p_2)>0$ so that if $p le p_2$ and the number of occupied vertices in the initial configuration is at most $epsilon_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2) log(n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and $epsilon_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.
We prove that if $pgeq 1$ and $-1leq rleq p-1$ then the binomial sequence $binom{np+r}{n}$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $ u(p,r)$, whose support is contained in $left[0,p^p(p-1)^{1-p}right]$. If $p>1$ is a rational number and $-1<rleq p-1$ then $ u(p,r)$ is absolutely continuous and its density function $V_{p,r}$ can be expressed in terms of the Meijer $G$-function. In particular cases $V_{p,r}$ is an elementary function. We show that for $p>1$ the measures $ u(p,-1)$ and $ u(p,0)$ are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence $binom{np+r}{n}$ is positive definite if and only if either $pgeq 1$, $-1leq rleq p-1$ or $pleq 0$, $p-1leq r leq 0$. The measures corresponding to the latter case are reflections of the former ones.
Chase-escape is a competitive growth process in which red particles spread to adjacent uncolored sites, while blue particles overtake adjacent red particles. We introduce the variant in which red particles die and describe the phase diagram for the resulting process on infinite d-ary trees. A novel connection to weighted Catalan numbers makes it possible to characterize the critical behavior.
Many authors have studied the phenomenon of typically Gaussian marginals of high-dimensional random vectors; e.g., for a probability measure on $R^d$, under mild conditions, most one-dimensional marginals are approximately Gaussian if $d$ is large. In earlier work, the author used entropy techniques and Steins method to show that this phenomenon persists in the bounded-Lipschitz distance for $k$-dimensional marginals of $d$-dimensional distributions, if $k=o(sqrt{log(d)})$. In this paper, a somewhat different approach is used to show that the phenomenon persists if $k<frac{2log(d)}{log(log(d))}$, and that this estimate is best possible.
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