How far to the nearest facility?
2nd October 2023
For planning the road network, and provision of services, it is valuable to know the distance from a typical point to the nearest facility. Examples include electric vehicles requiring a charging station or pedestrians requiring a taxi. Modelling the network and facilities as a Manhattan Poisson line Cox process, and using path distance (that is, along roads) we can find the distribution of the nearest facility from a typical point, and the distribution of the "k"th nearest facility from a typical intersection. Numerical simulations confirm the expressions obtained, as well as the improvement in accuracy compared with simpler models.
Survival and complete convergence for a branching annihilating random walk
9th October 2023
We study a branching annihilating random walk (BARW) in which particles move on the discrete lattice in discrete generations. Each particle produces a poissonian number of offspring which independently move to a uniformly chosen site within a fixed distance from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. This can be thought of as a very strong form of local competition and implies that the system is not monotone. For certain ranges of the parameters of the model we show that the system dies out almost surely or survives with positive probability. In an even more restricted parameter range we strengthen the survival results to complete convergence with a non-trivial invariant measure. A central tool in the proof is comparison with oriented percolation on a coarse-grained level, using carefully tuned density profiles which expand in time and are reminiscent of discrete travelling wave solutions. Based on a joint work with Matthias Birkner (JGU Mainz), Jiří Černý (University of Basel), Nina Gantert (TU Munich) and Pascal Oswald (University of Basel).
Schramm-Loewner evolution, natural length, and variation
16th October 2023
Schramm-Loewner evolution (SLE) describes a family of random curves that have a central role in random planar geometry. They appear as scaling limits of statistical physics models and, due to their conformal symmetries, are intimately linked to Brownian motion, Gaussian free fields, Liouville quantum gravity, etc. Although SLE curves are fractal, one can associate to them a notion of length. I will review these constructions and present an alternative construction of the natural length in terms of variation regularity. If time permits, I will comment on other results regarding the regularity of SLE.
Fermionic Gaussian free field structure in the Abelian sandpile model and uniform spanning tree
23rd October 2023
In this talk, we will derive a "spin representation" for certain observables of the Uniform Spanning Tree (UST). This representation is given in terms of an algebra of polynomials called Grassmannian algebra. We use such methods to calculate the joint cumulants of the fields related to the UST in the hypercubic lattice, including the height one field of the Abelian Sandpile Model. We will also explore the scaling limits of these fields in dimensions greater than one. Finally, we demonstrate that our analysis holds for triangular lattice and discuss further universality results. This talk is based on a joint work with A. Cipriani, A. Rapoport, and W. Ruszel.
Coverage with frayed edges
30th October 2023
We will examine certain coverage events for the Boolean model in stochastic geometry. The Boolean model is the union of balls of radius r centred at the points of a random point process. When this point process is supported on a bounded region A, the existence of the boundary becomes important for questions of coverage (for which r does the union of balls cover all of A?) as well as percolation-type questions such as the existence of a path connecting distant points. Inside a convex polytope, there are a number of interesting phase transitions based on the geometry of the boundary.
We will look at a number of these questions, including recent work on a question about the topology of the covered region, and techniques used to work near the boundary. Based on joint work with Mathew Penrose and Xiaochuan Yang.
The environment seen from a geodesic in last-passage percolation, and the TASEP seen from a second-class particle
6th November 2023
We study directed last-passage percolation in Z2 with i.i.d. exponential weights. What does a geodesic path look like locally, and how do the weights on and nearby the geodesic behave? We show convergence of the distribution of the "environment" as seen from a typical point along the geodesic in a given direction, as its length goes to infinity. We describe the limiting distribution, and can calculate various quantities such as the density function of a typical weight, or the proportion of "corners" along the path. The analysis involves a link with the TASEP (totally asymmetric simple exclusion process) seen from an isolated second-class particle, and we obtain some convergence and ergodicity results for that process. The talk is based on joint work with Allan Sly and Lingfu Zhang: https://arxiv.org/abs/2106.05242.
The Brownian spatial coalescent
13th November 2023
A Brownian spatial coalescent is axiomatically defined as a Markov coalescent process on the d-dimensional torus in continuous space whose underlying spatial motion is Brownian in the following sense: backwards in time, lineages follow independent Brownian bridges along the branches of the coalescence tree, conditional on its topology and times and locations of merge events. We allow for the most general case of simultaneous multiple mergers, and show that the law of a Brownian spatial coalescent is characterised by a family of "transition measures", reminiscent of the transition rates in the non-spatial setting.
We prove that the subset of Brownian spatial coalescents that are sampling consistent, in a novel sense, stands in one-to-one correspondence with the well known class of (non-spatial) Ξ-coalescents, or Λ-coalescents if simultaneous mergers are not allowed. This gives rise to what we call the Brownian spatial Ξ (or Λ)-coalescent. They describe the genealogies of a class of forward in time population models with non-constant population density. For example, the genealogies of the well known Fleming-Viot superprocess are described by the Brownian spatial Kingman coalescent.
An overview of set and manifold estimation
20th November 2023
Set estimation consists in estimating the support S (or related quantities such as its volume or its perimeter) of an unknown density from a finite sample of points in RD.
The first part of the talk is dedicated to presenting the three main support estimators and their properties when the support is full dimensional.
In a second part of the talk we will study how we can estimate the lower dimensional support S.
Throughout the talk I will try to emphasize the links between statistics, probabilities and discrete geometry and present some related open problems.
Compact support versus instantaneous propagation for spatial branching processes
27th November 2023
This talk will provide an introduction and overview of certain questions concerning the supports of continuous state space spatial branching processes, in particular superprocesses* and solutions to certain related SPDE. I will begin by discussing two classical, contrasting results: the compact support property for super-Brownian motion and instantaneous propagation of supports for super-Levy processes with jump motion. I will then discuss related questions for solutions to stochastic heat equations and some proof techniques before concluding with more recent developments in the area.
*Superprocesses will be defined.
On fraudulent stochastic algorithms
4th December 2023
We introduce and analyse the almost sure convergence of a stochastic algorithm for the global minimization of Morse functions on compact Riemannian manifolds. This diffusion process is called fraudulent because it requires the knowledge of minimal value of the function. Its investigation is nevertheless important, since in particular it appears as the limit behavior of non-fraudulent and time-inhomogeneous swarm mean-field algorithms for global optimization or in stochastic gradient descent algorithms in overparametrized learning applications. The talk is based on collaborations with Benaïm, Bolte and Villeneuve.
Road layout in the KPZ class
11th December 2023
In this talk I will be after a model for road layouts. Imagine a Poisson process on the plane for start points of cars. Each car picks an independent random direction and goes straight that way for some distance. I will start with showing that the origin (my house, that is) will see a lot of car traffic within an arbitrary small distance.
Which is not what we find in the real world out there, why? The answer is of course coalescence of paths in the random environment provided by hills and other geographic or societal obstacles. This points us towards first passage percolation (FPP), expected to be a member of the KPZ universality class. Due to lack of results for FPP, instead we built our model in exponential last passage percolation (LPP), known to be in the KPZ class. I will introduce LPP, then explain how to construct our road layout model in LPP, and what phenomena we can prove about roads and cars in this model using results from the LPP literature.