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Multiple scales in metapopulations can give rise to paradoxical behaviour: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilised in the well-mixed limit…
Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on…
The state of many physical, biological and socio-technical systems evolves by combining smooth local transitions and abrupt resetting events to a set of reference values. The inclusion of the resetting mechanism not only provides the…
A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit $\textit{metastable mixing}$ with $\textit{modes}$ $S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the…
In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect…
We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results established in [2]. First of all the random walk is transient in dimension at least three. Focusing on dimension two,…
A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the…
We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos and den Hollander, dos…
By generating transient encounters between individuals beyond their immediate social environment, transport can have a profound impact on the spreading of an epidemic. In this work, we consider epidemic dynamics in the presence of the…
Random walk methods are used to calculate the moments of negative image equilibrium distributions in synaptic weight dynamics governed by spike-timing dependent plasticity (STDP). The neural architecture of the model is based on the…
We consider a quantized version of the Sinai-Derrida model for "random walk in random environment". The model is defined in terms of a Lindblad master equation. For a ring geometry (a chain with periodic boundary condition) it features a…
Many systems across the sciences evolve through a combination of multiplicative growth and diffusive transport. In the presence of disorder, these systems tend to form localized structures which alternate between long periods of relative…
Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning…
We derive the characteristic function of stochastic functionals of a random walk whose position is reset to the origin at random times drawn from a general probability distribution. We analyze the long-time behavior and obtain the temporal…
The task of \emph{community detection} in a graph formalizes the intuitive task of grouping together subsets of vertices such that vertices within clusters are connected tighter than those in disparate clusters. This paper approaches…
The diffusion of a particle in a crowded environment typically proceeds through three regimes: for very short times the particle diffuses freely until it collides with an obstacle for the first time, while for very long times diffusion the…
The rotor-router model is a deterministic process analogous to a simple random walk on a graph. This paper is concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational…
We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -\alpha n$…
Critical behaviour of a fluid, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group. As a simplified model, relaxational stochastic dynamics of a non-conserved scalar order…
Positive density-dependence occurs when individuals experience increased survivorship, growth, or reproduction with increased population densities. Mechanisms leading to these positive relationships include mate limitation, saturating…