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Register a new userPhase transitions in non-linear urns with interacting types
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We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three phases: as well as a phase with an almost sure limit where each of the three colours is equally represented and a phase with almost sure convergence to an asymmetric limit, which both occur with two types, there is also an intermediate phase where both symmetric and asymmetric limits are possible. In a model with anti-symmetric interactions between the types, we show the existence of a phase where the proportions of the three colours cycle and do not converge to a limit, alongside a phase where the proportions of the three colours can converge to a limit where each of the three is equally represented.
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A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). We show that, with probability one, eventually only a random subset of non-interacting components of the process survives. A similar result also holds for the relevant generalized Polya urn model with removals.
We study survival among two competing types in two settings: a planar growth model related to two-neighbour bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncoloured sites are given a colour at rate $0$, $1$ or $infty$, depending on whether they have zero, one, or at least two neighbours of that colour. In the urn scheme, each vertex of a graph $G$ has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same colour is added to each neighbouring urn, and balls in the same urn but of different colours annihilate on a one-for-one basis. We show that, for every connected graph $G$ and every initial configuration, only one colour survives almost surely. As a corollary, we deduce that in the two-type growth model on $mathbb{Z}^2$, one of the colours only infects a finite number of sites with probability one. We also discuss generalisations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.
An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0leq L<Uleq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_noverset{a.s.}longrightarrow Z$ for some random variable $Z$, and begin{gather*} D_n:=sqrt{n},(Z_n-Z)longrightarrowmathcal{N}(0,sigma^2)quadtext{conditionally a.s.} end{gather*} where $sigma^2$ is a certain random variance. Almost sure conditional convergence means that begin{gather*} Pbigl(D_nincdotmidmathcal{G}_nbigr)overset{weakly}longrightarrowmathcal{N}(0,,sigma^2)quadtext{a.s.} end{gather*} where $Pbigl(D_nincdotmidmathcal{G}_nbigr)$ is a regular version of the conditional distribution of $D_n$ given the past $mathcal{G}_n$. Thus, in particular, one obtains $D_nlongrightarrowmathcal{N}(0,sigma^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.
We consider the preferential attachment model with multiple vertex types introduced by Antunovic, Mossel and Racz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunovic, Mossel and Racz. We also consider another family of examples where we show that the conjecture does hold.
We consider a simple discrete-time Markov chain with values in $[0,infty)^{Z^d}$. The Markov chain describes various interesting examples such as oriented percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the total number of particles in this framework. The main results are roughly as follows: If $d ge 3$ and the Markov chain is not too random, then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, $d=1,2$, or the Markov chain is random enough, then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the Markov chain with proper normalization.
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