Consider a biased random walk in positive random conductances on ℤ^d for d ≥ 5. In the sub-ballistic regime, Fribergh and Kious (2018) proved the convergence, under the annealed law, of the properly rescaled random walk towards a Fractional Kinetics. I will explain that a quenched equivalent of this theorem is true and a strategy to simplify the question. This is a joint work with A. Fribergh and T. Lions.

Location: Room 8W 2.28, 8 West, University of Bath

Time: 22 February 2023, 2:15pm GMT

Random geometric graphs are a large and diverse family of graphs with various interesting properties. The simplest type is the Poisson Boolean model. The vertices are given by a Poisson point process and we create an edge for any two points that are sufficiently close. It is driven by a single parameter, the Poisson point process intensity. The main object of interest is the phase transition of the graph, driven by the intensity which divides the parameter space into two regions of vastly different behaviours. One particular change is the size of large components around the origin.

Location: Room 8W 2.34, 8 West, University of Bath

Time: 30 November 2022, 1:15pm GMT

Biomimicry is the study of biology in search of natural solutions to our problems and has been around for over 4000 years. One of the most recent developments in this field is the slime mold Physarum polycephalum and its ability to solve network problems without centralised intelligence. This talk aims to understand this phenomenon through reinforced random walks on graphs.

Location: Room G.12, Fry Building, University of Bristol

Time: 27 October 2022, 4:00pm GMT

During evolution, a population can acquire and accumulate mutations, i.e. changes in the DNA sequence which are transmitted across generations. Importantly, mutations can affect the ability of a living organism to survive and reproduce. This ability is called fitness, and it is expected that deleterious mutations (i.e. mutations that decrease fitness) have a lower probability of being transmitted to succeeding generations. In spatially structured populations, however, the probability an individual survives depends not only on the fitness of its genotype, but also on the local population density in the neighbourhood of this individual. Roughly speaking, a high local population density may either increase the survival probability due to cooperativity or decrease it due to competition. This interaction may increase the probability of mutations arising in the front of a spatial population expansion to spread over a large unoccupied habitat, a phenomenon called gene surfing. In this talk, I will explain a model introduced by Foutel-Rodier and Etheridge in 2020 to study the gene surfing of deleterious mutations, as well as a conjecture regarding the scaling limit of the model. If time allows, I will also show some of the techniques that we are planning to use in a birth-death toy model with countably many types of particles.

Location: Room G.12, Fry Building, University of Bristol

Time: 27 October 2022, 4:00pm GMT

The symbiotic contact process can be thought of as a two type generalisation of the contact process which can be used to model the spread of two symbiotic diseases. Each site can either be infected with type A, type B, both, or neither. Infections of either type at a given site occur at a rate of lambda multiplied by the number of neighbours infected by that type. Recoveries of either type at a given site occur at rate 1 if only one type is present, or at a lower rate mu if both types are present, hence the symbiotic name. Both the contact process and the symbiotic contact process have two critical infection rates on a Galton-Watson tree, one determining weak survival, and the other strong survival. Here, weak survival refers to the event where at least one A infection and at least one B infection is present at all times. Strong survival is the event that the root of the tree is infected with both A and B infections at the same time infinitely often. In this talk, I will prove that for small values of mu the weak critical infection rate for the symbiotic model is strictly smaller than the critical rate for the contact process. I will also discuss the more complicated case of strong survival for both processes.

Location: Room 2.32, Fry Building, University of Bristol

Time: 31 August 2022, 4:00pm GMT

On the lattice ℤ², consider coalescing random walks starting at every point, which move up or right with equal probability, forming the edges of a graph - the discrete web. The resulting connected graph can be seen as a collection of one-dimensional coalescing random walks, where one diagonal axis represents space and the other time. Inspired by the voter model, known to be dual to coalescing random walks, this object yields, via diffusive space-time scaling limit, an uncountable analogue, the Brownian web, of one-dimensional coalescing Brownian motions starting at every point in 2D spacetime. We consider the random walk on the discrete web, many of whose properties we can understand by studying the growth of graph-balls and effective resistance.

Location: Room 8W 2.27, 8 West, University of Bath

Time: 27 July 2022, 1:15pm GMT

We introduce reinforced digging random walks (RDRW) which are self-interacting non-Markovian random walks that depend on the reinforcement parameter ẟ > 0 and some reinforcement function f. RDRW can be seen as a mixture of a digging random walk and a random walk with random conductances. In particular, we study RDRW with a linear reinforcement and show that the phase transition for recurrence and transience depends on the branching-ruin number of a tree and some quantity K(ẟ).

Location: Room 2.32, Fry Building, University of Bristol

Time: 22 June 2022, 12:00pm GMT

Consider a mean field Erdős-Rényi random digraph process on n vertices. Let each possible directed edge arrive with rate 1/2n. Without opposition, this process is guaranteed to result in the n-complete graph. Hence, we introduce a Poisson rain of "lightning strikes" to each vertex with rate λ(n), which propagates across its out-graph, burning all edges of an affected vertex. Subsequently, the system continues to fluctuate to no end in a battle between creation and destruction.

Location: Wolfson Lecture Theatre (4W 1.7), 4 West, University of Bath

Time: 25 May 2022, 1:15pm GMT