Probability in the North East Workshop
9 April 2026
Organisers: Irene Ayuso Ventura and Oliver Tough
Venue: Durham University, Math Department building, Scott Logic Lecture Theater
Attendence is free but registration is required for organisational purposes,
by filling this form
by 27 March 2026. Limited travel funding will be available, with priority given to PhD students
and postdoctoral researchers.
We invite early career researchers to apply for contributed talks!
This event is supported by an LMS Scheme 3 grant, and Additional Funding Programme
for Mathematical Sciences, delivered by (EPSRC EP/V521917/1 - Heilbronn Institute)
and (EP/V521929/1 - INI).
Programme
10:15-11:00
Welcome!
11:00-11:45
Francesca Cottini
(Sorbonne Université)
Directed polymers in random environments describe a perturbation of the simple random walk given by a random disorder (environment). The partition functions of this model have been thoroughly investigated in recent years, also motivated by their link with the solution of the Stochastic Heat Equation. While classical results focus on space-time independent disorder, we consider a Gaussian environment with (critical) spatial correlations decaying as $|x|^{-2}$ times a slowly varying function. We show that a phase transition, analogous to that in the space-time independent case, still occurs: in the high temperature regime the log-partition function satisfies a central limit theorem, while it vanishes in law in the low temperature regime. Remarkably, the inverse temperature needs to be tuned differently from the independent case, where the scaling constant $\hat{\beta}$ emerges from a nontrivial multi-scale dependence in the second moment computation — the core technical challenge of the work. Based on a joint work with Clément Cosco (Paris Dauphine) and Anna Donadini (Milano-Bicocca).
11:45-12:30
Bas Lodewijks
(University of Sheffield)
We study a random tree model known as the Preferential Attachment tree with Vertex Death. Here, one can both add vertices to the tree as well as kill vertices. This model mimics the non-monotone growth of real-world networks, absent in classical preferential attachment models. One initialises the tree with a single root vertex labelled 1. At every step $n$, either a new vertex labelled $n+1$ is added to the tree and connected to an already present alive vertex, selected preferentially according to a function $b$, or an already present vertex is selected preferentially according to a function $d$ and killed. Killed vertices can make no new connections. We are interested in the behaviour of the alive vertex $I_n$ with the largest degree (the richest alive vertex) and the alive vertex $O_n$ with the smallest label (the oldest alive vertex). In this talk we focus on two distinct regimes in which behaviour is different: (i) The Rich Are Old regime, where we provide conditions under which I_n/O_n is tight and conditions under which I_n/O_n tends to infinity. (ii) The Rich Die Young regime, where I_n/O_n always tends to infinity. We shall discuss how the two regimes can be identified and what drives the behaviour observed in each regime. Partially joint work with Markus Heydenreich.
12:30-14:00
Lunch
14:00-14:30
Maxence Baccara
École Polytechnique
Motivated by questions in population ecology, we study a toy model in which $N$ heavy-tailed random walks in $\ZZ^d$ consume resources present at the initial time. Our objective is then to establish a law of large numbers and a central limit theorem for the amount of resources consumed by one of the random walks. We will compare the result obtained with the classical result of Le Gall-Rosen ('91), which corresponds to the case $N=1$.
14:30-14:45
(Mini)break
14:45-15:30
Pascal Maillard
(Université Toulouse III)
The continuous random energy model (CREM) is a certain Gaussian process indexed by a binary tree of depth \(T\). It was introduced by Derrida and Spohn in and by Bovier and Kurkova in as a toy model of a spin glass.
In this talk, I will present recent results on hardness thresholds for algorithms that search for low-energy states. I will first discuss the existence of an algorithmic hardness threshold \(x_*\): finding a state of energy lower than \(-x T\) is possible in polynomial time if \(x < x_*\), and takes exponential time if \(x > x_*\), with high probability. I will also discuss related results on the complexity of sampling the Gibbs measure of inverse temperature parameter \(\beta>0\).
I shall then focus on the transition from polynomial to exponential complexity near the algorithmic hardness threshold of the optimization problem, by studying the performance of a certain beam-search algorithm of beam width \(N\) depending on \(T\) --- we believe this algorithm to be natural and asymptotically optimal. The algorithm turns out to be essentially equivalent to the time-inhomogeneous version of the so-called \(N\)-particle branching Brownian motion (\(N\)-BBM), which has seen a lot of interest in the last two decades. Studying the performance of the algorithm then amounts to investigating the maximal displacement at time \(T\) of the time-inhomogeneous \(N\)-BBM. In doing so, we are able to quantify precisely the nature of the transition from polynomial to exponential complexity, proving that the transition happens when the log-complexity is of the order of~\(T^{1/3}\). This result appears to be the first of its kind and we believe this phenomenon to be universal in a certain sense.
Based on joint works with Louigi Addario-Berry, Fu-Hsuan Ho and Alexandre Legrand, respectively.
15:30-16:00
Break
16:00-16:45
Anne Schreuder
(Aalto University)
Since Charles Loewner’s seminal paper in 1923 Loewner chains have been a powerful tool in Mathematics. Originally, applied to distortion estimates for univalent functions such as Bieberbach’s conjecture/de Branges theorem, Loewner chains have been rediscovered as a method to describe random curves, e.g. Schramm-Loewner Evolutions, and random growth models such as Diffusion Limited Aggregation (DLA) or Hasting-Levitov type growth models. This works as by the Riemann Mapping Theorem there is a bijection between Loewner chains and (continuously) growing simply connected sets in the plane. Crucially, there is also a bijection between Loewner chains and finite Borel measures which constitutes an analytic representation theorem. One direction of this bijection is given by the famous Loewner-Kufarev Equation. We were interested in the converse question: How can the driving measure be obtained from its Loewner chain. In particular, we obtain explicit formulations of the driving measure and an expression for the density of the driving measure.