Probability seminar abstracts

Bruno Schapira

Sharp asymptotics for the expected number of pioneers in high dimensional percolation

28th September 2026
Consider critical percolation in dimension d>6, under the usual assumption that the two-point function behaves as the Green's function of simple random walk. In this setting, we show that given any set S containing the origin, the expected number of points on the boundary S which are connected to the origin by an open path that stays in S is of the same order as the probability that a simple random walk exits the set S. In particular it is bounded from above by a universal constant, a fact which outside the case of half-spaces (and the trivial case of finite sets), is new. This result has several interesting consequences, which we will discuss. Based on joint work with Romain Panis, and ongoing work with Amine Asselah and Perla Sousi.

Emma Bailey

Conditional and unconditional results for extremal values of zeta

5th October 2026
The Riemann zeta function encodes information about primes and its behaviour, particularly along the line 1/2+it, is of central interest. If one takes the 'height', t, uniformly at random, then the function behaves log-normally. I鈥檒l present results, conditional and unconditional on the Riemann Hypothesis, about large deviations of this random variable. No number theoretic knowledge will be assumed. This work is joint with L-P Arguin and Asher Roberts.

Renan Gross

Polygonal side gluings: mixing, combinatorics, and (hyperbolic) geometry

12th October 2026
Take a polygon with 2n sides, and glue the sides together according to some matching. Congratulations! You now have a compact surface, whose genus g depends on the details of the gluing. In this talk, we discuss some fun things you can do with such gluings, focusing on a natural Markov chain which generalizes the planar chord swap, a not-yet-tightly-solved chain on Catalan structures.

Joint work with An膽ela 艩arkovi膰.

Delphin S茅nizergues

The height of the infection tree

19th October 2026
We consider an SIR (Susceptible-Infected-Removed) epidemic model started with one infected individual in a population containing a large number n of healthy individuals. Each individual recovers at rate 1 and infects each healthy individual at rate 位_n. We encode all the infection events that take place during the epidemic in a tree T_n. We are interested in the height of this "infection tree" T_n, in particular in the regime where 位_n=位/n with 位 a constant greater than 1, which requires a specific analysis. In this regime, the asymptotic height of the tree exhibits a phase transition in 位 around a value 位_c 鈮� 1.80. I will explain the main steps of the proof, which include a comparison with Bienaym茅-Galton-Watson trees in the subcritical phase, and a precise control of the profile of these trees in the supercritical phase.

Joint work with Emmanuel Kammerer and Igor Kortchemski.

Quentin Berger

2D directed polymers and the Stochastic Heat Flow

26th October 2026
I will review some recent results on the directed polymer model in dimension d=2, which can be seen as a discretised version of the stochastic heat equation. Recent results have shown that, in a suitable (and subtle) critical window of parameters, the model converges to a disordered limit, namely the Critical 2D Stochastic Heat Flow (SHF) introduced by Caravenna, Sun and Zygouras. This SHF is a non-Gaussian stochastic process of measures on R^2, and a lot of its properties have been studied over the last years. In this talk, I will first take some time to explain the (idea of the) construction of the SHF and then review some of the recent advances. In particular, I will comment on recent works in collaboration with F. Caravenna, N. Turchi and N. Zygouras, which study how the mass assigned by the SHF to a ball behaves, either as time goes to infinity or as the radius of the ball goes to 0.