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Moment asymptotics for branching random walks in random environment
We consider the long-time behaviour of a branching random walk in random environment on the lattice Zd. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments m_np , i.e., the p-th moments over the medium of the n-th moment over the migration and killing/branching, of the local and global population sizes. For n = 1, this is well-understood citeGM98, as m_1 is closely connected with the parabolic Anderson model. For some special distributions, citeA00 extended this to ngeq2, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for m_n. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that m_n^p m_1^np are asymptotically equal, up to an error e^o(t). The cornerstone of our method is a direct Feynman-Kac-type formula for mn, which we establish using the spine techniques developed in citeHR1.1
Moment asymptotics for multitype branching random walks in random environment
We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/pij for the upper tail of the mean number of j type particles produced by an i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.
Extremal aging for trap models
In the seminal work [5], Ben Arous and Cerný give a general characterization of aging for trap models in terms of α-stable subordinators with α ∈ (0,1). Some of the important examples that fall into this universality class are Random Hopping Time (RHT) dynamics of Random Energy Model (REM) and p-spin models observed on exponential time scales. In this paper, we explain a different aging mechanism in terms of extremal processes that can be seen as the extension of α-stable aging to the case α=0. We apply this mechanism to the RHT dynamics of the REM for a wide range of temperature and time scales. The other examples that exhibit extremal aging include the Sherrington Kirkpatrick (SK) model and p-spin models [6, 9], and biased random walk on critical Galton-Watson trees conditioned to survive [11].
Parabolic Anderson model with a finite number of moving catalysts
We consider the parabolic Anderson model (PAM) which is given by the equation partial u/partial t = kappaDelta u + xi u with ucolon, Z^dtimes [0,infty)to R, where kappa in [0,infty) is the diffusion constant, Delta is the discrete Laplacian, and xicolon,Z^dtimes [0,infty)toR is a space-time random environment. The solution of this equation describes the evolution of the density u of a ``reactant'' u under the influence of a ``catalyst'' xi.newlineindent In the present paper we focus on the case where xi is a system of n independent simple random walks each with step rate 2drho and starting from the origin. We study the emphannealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. xi and show that these exponents, as a function of the diffusion constant kappa and the rate constant rho, behave differently depending on the dimension d. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents,...
Fluid and diffusion limits for the Poisson encounter-mating model
Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this article we study SEM with Poisson firing times. We prove that an infinite population corresponds to a fluid limit, i.e., the stochastic dynamics converges to a deterministic system governed by coupled ODEs. Moreover, we establish a functional central limit theorem and give a diffusion approximation for the model. Next, we convert the fluid limit ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Under the so-called fine balance condition, which characterizes panmixia for finite populations, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of females and males. Without the fine balance assumption, but under some symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to fully characterize assortative mating.
The stochastic encounter-mating model
We propose a new model of permanent monogamous pair formation in zoological populations comprised of kge 2 types of females and males, which unifies and generalizes the encounter-mating models of Gimelfarb (1988). In our model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, which depend on the sex and the type of the animals, we analyze the contingency table Q(t) of permanent pair types at any time tge 0. First, we consider definite mating upon encounter and provide a formula for the distribution of Q(t). In particular, at the terminal time T, the so-called mating pattern Q(T) has a multiple hypergeometric distribution. This implies panmixia which means that female and male types are uncorrelated in the expected mating pattern. Next, when the firing times come from Poisson and Bernoulli point processes, we formulate conditions that characterize panmixia. Moreover, when these conditions are satisfied, the underlying parameters of the model can be changed to yield definite mating upon encounter, and our results for the latter case carry over. Finally, when k=2, we fully characterize heterogamy/panmixia/homogamy, i.e., negative/zero/positive correlation of same type females and males in the expected mating pattern. We thereby rigorously prove, strengthen and generalize Gimelfarb's results.
The parabolic Anderson model on the hypercube
We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and the k-th largest potential. One of our main motivations in this article is to investigate the mutationselection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.
Aging in the GREM-like trap model
The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume L-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the two-time correlation function of the infinite volume L-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any L, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.