The Department of Mathematics at the University of Manchester will host the UK Easter Probability Meeting, a week-long workshop on probability and its applications, with three minicourses, 12 invited talks and opportunities for contributed talks from early-career researchers.
In this course, we shall consider a model for evolution in a spatial continuum, called the spatial Lambda-Fleming-Viot process, which was introduced by Alison Etheridge (Univ. of Oxford) and Nick Barton (IST Vienna) in 2008. We shall discuss its dual "genealogical" process and give examples of interesting behaviour of the local allele frequencies under different parameter regimes and on different space and time scales.
We study the 2d Stochastic Heat Equation, that is the heat equation in two space dimensions with a multiplicative random potential (space-time white noise). This equation is ill-defined due to the singularity of the potential and we regularise it by discretising space-time. We prove that, in the limit when discretisation is removed and the noise strength is rescaled in a critical way, the solution has a well-defined and unique limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow. We investigate its features, showing that it cannot be the exponential of a generalised Gaussian field.
(joint work with R. Sun and N. Zygouras)
Exponential functionals of subordinators have been thoroughly investigated, since they play a key role in several facets of modern probabilities, as they correspond to the extinction times of non-increasing self-similar Markov processes. As such, they are involved in the description of various processes ranging from the analysis of algorithms to coagulation or fragmentation processes. In this talk we will provide the exact large-time equivalents of the density and upper tail distribution of the exponential functional of a subordinator in terms of its Laplace exponents. This improves previous results on the logarithmic asymptotic behaviour of this tail.
We will then see how this result can be used to determine the large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity. The extinction time of a typical fragment in such a process is an exponential functional of a subordinator. But the tail of the extinction time of the whole fragmentation process decreases much more slowly in general. We will quantify this difference by determining the asymptotic ratio of the two tails.
The N-particle branching random walk is a discrete time branching particle system with selection consisting of N particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant.
I will discuss recent results and open conjectures about the long-term behaviour of this particle system when N, the number of particles, is large. In the case where the jump distribution has regularly varying tails, building on earlier work of J. Bérard and P. Maillard, we prove that at a typical large time the genealogy is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.
Based on joint work with Matt Roberts and Zsófia Talyigás.
I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form.
This talk is based on joint work with Nicholas Georgiou (Durham).
A Pólya urn is a Markov process describing the contents of an urn that contains balls of d colours. At every time step, we draw a ball uniformly for the urn, note its colour, then put it back in the urn along with a set of new balls which depend on the colour drawn. The number of balls of colour j added when colour i is drawn is given by the (i,j)th entry of a predetermined replacement matrix R. The asymptotic behaviour of the urn for most replacement matrices can be inferred from the following two canonical cases. In the case of R the identity, the proportion of each colour in the urn tends to a Dirichlet distributed random variable with parameter given by the urn's initial composition. In the case of R irreducible, this limit is a deterministic vector that only depends on R. Fluctuations around these limits are also known.
Recently, Borovkov showed results on the asymptotic behaviour in the identity case, when the initial number of balls grows together with the number of time steps. In this talk, we show analogous results for the irreducible case. This will include the asymptotic behaviour of the proportion of each colour in the urn and the fluctuations around this limit.
The meeting will take place in the Department of Mathematics at the University of Manchester.
Alan Turing Building is building 46 on the Campus Map. For more details on reaching the venue, please see Maps and Travel.
The meeting schedule appears as an embedded Google Calendar below. You may need to set your adblocker to allow iframes to see it. Alternatively, you can access it at this link: UK Easter Probability Meeting 2023 schedule.
Registration is now open. The registration deadline is Wednesday 15 March. There is a registration fee of £50. After you have registered at Eventbrite, we will contact you to arrange payment.
There will be a conference dinner on Wednesday at no additional charge.
There is an opportunity for postdocs and UK research students to present a short talk. Space for the short talks is limited, and participants will be selected based on their proposed abstracts. The deadline for abstract submission is Wednesday 16 February, and decisions will be communicated by Monday 20 February. Participants selected to present a short talk will receive financial support for their attendance and have the conference fee waived. There is also an opportunity for all participants to present a poster.
Funding is available to support the attendance of UK research students, with priority for EPSRC-funded students, and to pay for caring costs of participants. We expect to confirm to participants whether we are able to offer funding shortly after registering. When there is no further funding available, this will be communicated clearly here.
The organising committee is:
If you have any questions, please email the committee.
This meeting is funded by grants from the London Mathematical Society, EPSRC and the Applied Probability Trust.