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’ll 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 Anđela Šarković.
Delphin Sénizergues
Largest positive induced subtree in a large uniform p-signed binary tree
19th October 2026
For a given integer n, we consider a tree T_n chosen uniformly at random among binary trees with n leaves, and we label each of its internal nodes with a sign +/−, chosen i.i.d. with probabilities p / (1−p). We are interested in the associated random variable X_n, defined as the size of the largest subset of leaves that induces in T_n a subtree carrying only + signs on its nodes. The original problem stems from the fact that the tree T_n actually encodes a random separable permutation, and X_n is then the length of the longest increasing subsequence of this permutation. We will show that, as n grows large, X_n behaves like nα for some exponent α=α(p), and we will discuss the value of α(p) in more detail.
(Joint work with Arka Adhikari, Jacopo Borga, Thomas Budzinski and William Da Silva)
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 R2, 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.
Daniel de la Riva Massaad
Continuity of the phase transition for long-range Bernoulli percolation
2nd November 2026
We prove continuity of the phase transition for long-range Bernoulli percolation on Z2 with connection probabilities that decay at least exponentially fast. This extends earlier work by Duminil-Copin, Sidoravicius, and Tassion on slabs. Our proof establishes a finite-size criterion that allows us to run a renormalisation argument. As a corollary, we obtain supercritical sharpness, in the sense that for every β>β_c, there exists a unique macroscopic cluster in large boxes with high probability.
Philip Easo
Supercritical sharpness of percolation
9th November 2026
Given an infinite transitive graph (such as the lattice Zd), consider the random subgraph obtained by independently retaining each edge with probability p. This model undergoes a phase transition as p varies across a critical value p_c that marks the emergence of an infinite component. A classical result states that for each p < p_c, the probability that a given vertex belongs to a component of size at least n decays exponentially in n. Sahar Diskin, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion, and I have recently proved the analogous result for p > p_c.