# Probability seminar abstracts

Our seminars are usually held on Mondays at 1.15pm. In normal times they are in room 4W 1.7, but in 2020-21 they will be held via Zoom.

# Amine Asselah

**Large deviation for the intersection of the trace of 2 random walks**

*5th October 2020*

We discuss a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander. We obtain also some path properties.

(Joint work with B. Schapira.)

# Benjamin Fehrman

**Large-deviations for conservative, stochastic PDE and non-equilibrium fluctuations**

*19th October 2020*

The large deviations of certain interacting particle systems about their hydrodynamic limits have long been formally connected to the small-noise large deviations of certain stochastic PDEs with conservative noise. This relationship had, however, remained non-rigorous due to the fact that such SPDEs are degenerate and supercritical, and therefore lack a suitable solution theory. In this talk, I will explain the well-posedness of a sequence of approximating SPDEs, and I will show that there exists a scaling regime for which the solutions of the approximating equations satisfy a genuine large deviations principle equal to that of the particle process. The results are based on a detailed analysis of the associated skeleton equation - a degenerate parabolic-hyperbolic PDE - in energy critical spaces.

# Laure Dumaz

**Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization**

*2nd November 2020*

We consider the continuous Schrödinger operator - d^{2} / dx^{2} + B’(x) on the interval [0,L] where the potential B’ is a white noise. We study the spectrum of this operator in the large L limit. We show the convergence of the smallest eigenvalues and of the eigenvalues in the bulk towards a Poisson point process, and the localization of the associated eigenvectors in a precise sense. We also find that the transition towards delocalization holds for large eigenvalues of order L, where the limiting law of the point process corresponds to Sch_tau, a process introduced by Kritchevski, Valko and Virag for discrete Schrodinger operators. In this case, the eigenvectors behave like the exponential Brownian motion plus a drift, which proves a conjecture of Rifkind and Virag. Moreover the rescaled and unitarily-changed operator converges (in the strong resolvent sense) towards another differential operator, acting on R^{2} -valued functions and of the form "J\partial_t + 2*2 noise matrix" (where J is the matrix ((0, -1)(1, 0))). Joint works with Cyril Labbé.

# Gonçalo dos Reis

**Large deviations and exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts**

*16th November 2018*

We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is constrained to a (not necessarily bounded) convex domain by a local time on the boundary. These equations include the subclass of reflected self-stabilizing diffusions that drift towards their mean via convolution of the solution law with a stabilizing potential.
Firstly, we establish existence and uniqueness results for this class and address the propagation of chaos. We work with a broad class of coefficients, including drift terms that are locally Lipschitz in spatial and measure variables. However, we do not rely on the boundedness of the domain or the coefficients to account for these non-linearities and instead use the self-stabilizing properties. We prove a Freidlin-Wentzell type Large Deviations Principle and an Eyring-Kramer's law for the exit-time from subdomains contained in the interior of the reflecting domain.

This is joint work with Daniel Adams, Romain Ravaille, William Salkeld and Julian Tugaut.

# András Tóbiás

**The interplay of dormancy and transfer in bacterial populations: Invasion, fixation and coexistence regimes**

*7th December 2020*

We investigate the interplay between two fundamental mechanisms of microbial population dynamics and evolution called dormancy and horizontal gene transfer. The corresponding traits come in many guises and are ubiquitous in microbial communities, affecting their dynamics in important ways. Recently, they have each moved (separately) into the focus of stochastic individual-based modelling (Billiard et al. 2016, 2018; Champagnat, Méléard and Tran, 2019; Blath and Tóbiás 2019). Here, we examine their combined effects in a unified model. Indeed, we consider the (idealized) scenario of two sub-populations, respectively carrying 'trait 1' and 'trait 2', where trait 1 individuals are able to switch (under competitive pressure) into a dormant state, and trait 2 individuals are able to execute horizontal gene transfer, turning trait 1 individuals into trait 2 ones, at a rate depending on the frequency of individuals.

In the large-population limit, we examine the fate of a single trait i mutant arriving in a trait j resident population living in equilibrium, for i,j=1,2, i \neq j. We provide a complete analysis of the invasion dynamics in all cases where the resident population is individually fit and the initial behaviour of the mutant population is non-critical. We identify parameter regimes for the invasion and fixation of the new trait, stable coexistence of the two traits, and 'founder control' (where the initial resident always dominates, irrespective of its trait). The most striking result is that stable coexistence occurs in certain scenarios even if trait 2 (which benefits from transfer at the cost of trait 1) would be unfit when being merely on its own. In the case of founder control, the limiting dynamical system has an unstable coexistence equilibrium. In all cases, we observe the classical (up to 3) phases of invasion dynamics à la Champagnat 2006. The subject of this talk is joint work with Jochen Blath.

# Ellie Archer

**Random walks on decorated Galton-Watson trees**

*11th January 2021*

Random trees are the building blocks for a range of probabilistic structures, including percolation clusters on the lattice and many statistical physics models on random planar maps. In this talk we consider a random walk on a critical "decorated" Galton-Watson tree, by which we mean that we first sample a critical Galton-Watson tree T, replace each vertex of degree n with an independent copy of a graph G(n), and then glue these inserted graphs along the tree structure of T. We will determine the random walk exponents for this decorated tree model in terms of the exponents for the underlying tree and the exponents for the inserted graphs. We will see that the model undergoes several phase transitions depending on how these exponents balance out.

# Wolfgang König

**A grid version of the interacting Bose gas**

*15th February 2021*

The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose--Einstein condensation phase transition is sought for. We introduce a simplified version of the model in Z^{d} instead of R^{d} and with an organisation of the particles in deterministic box-like grids instead of Brownian cycles as the marks of a reference Poisson point process. We derive an explicit and interpretable variational formula in the thermodynamic limit for the canonical ensemble for any value of the particle density. In this formula, each of the microscopic particles and the macroscopic part of the configuration are seen explicitly (if they exist); the latter receives the interpretation of the condensate. The methods comprises a two step large-deviation approach for marked Poisson point processes and an explicit distinction into microscopic and macroscopic marks. We discuss the condensate phase transition in terms of existence of minimizer.

(Based on joint works with Adams/Collevecchio (2011) and Jahnel (ongoing).)

# Mickaël Maazoun

**Scaling limits of Baxter permutations and bipolar orientations**

*22nd February 2021*

The theory of permutons allows us to express scaling limits of the diagram of permutations. Scaling limits of uniform elements in various classes of pattern-avoiding permutations have attracted a fair amount of attention lately. We show such a result for Baxter permutations, a famous class of permutations avoiding generalized patterns.

A remarkable bijection of Bousquet-Mélou, Bonichon and Fusy (2010) with bipolar orientations, a type of decorated planar maps, allows us to express a Baxter permutation in terms of the relationship between the mating-of-trees encoding (Kenyon, Miller, Sheffield, Wilson, 2015) of a bipolar orientation and the one of its dual map. This was already studied by Gwynne, Holden, Sun (2016), and our main result can be seen as an improvement of theirs.

The main step of our approach is to encode the problem in a "coalescent-walk" process, which converges to the coalescing process obtained when solving the perturbed Tanaka SDE (Prokaj, 2011) with the same Brownian noise at different starting times. If time allows, I will talk about the robustness of the method and possible generalizations.

# Verónica Miró Pina

**Xi-coalescents arising in population models with bottlenecks**

*1st March 2021*

In this work, we study a family of Xi-coalecents that arise from a class of Wright-Fisher models with recurrent demographic bottlenecks. This family of Xi-coalescents constructed from i.i.d mass, is an extension of the symmetric coalescent (Gonzalez Casanova et al. 2021). We study the process which counts the frequency of blocks of each size in a n-coalescent, when n tends to infinity.

We show that the multivariate Lamperti transform of this process is a Markov Additive Process (MAP). This allows us to provide some asymptotics for the length of order r in a n-coalescent, which is defined as the sum of the lengths of all the branches that carry a subtree with r leaves.

Joint work with: Adrián González Casanova, Arno Siri-Jégousse and Emmanuel Schertzer.

# Paul Jenkins

**Asymptotic genealogies of interacting particle systems**

*15th March 2021*

Interacting particle systems are a broad class of stochastic models for phenomena in disciplines including physics, engineering, biology, and finance. A prominent class of such models can be expressed as a sequential Monte Carlo algorithm in which the aim is to construct an empirical approximation to a sequence of measures. The approximation is constructed by evolving a discrete-time, weighted population of particles, alternating between a Markov update and a resampling step. In this talk I discuss how to characterise the genealogy underlying this evolving particle system. More precisely, under certain conditions we can show that the genealogy converges (as the number of particles grows) to Kingman's coalescent, a stochastic tree-valued process widely studied in population genetics. This makes explicit the analogy between sequential Monte Carlo and an evolving biological population. This is joint work with Suzie Brown, Adam Johansen, Jere Koskela, and Dario Spanò.

# Leo Rolla

**Local and global behavior of the subcritical contact process**

*19th April 2021*

We will describe the scaling limit of the subcritical contact process in terms of a marked Poisson point process and a quasi-stationary distribution, and discuss the question of uniqueness of the QSD in this and other contexts. Based on joint works with E. Andjel, F. Ezanno and P. Groisman, with A. Deshayes, and with F. Arrejoría and P. Groisman.

# Delphin Sénizergues

**Asymptotic expansion for the height of weighted recursive trees**

*26th April 2021* **at 16:15**

Weighted recursive trees (WRTs) are built by successively adding vertices with predetermined weights to a tree: each new vertex is attached to a parent that is chosen among the vertices already present with probability proportional to their weights. I will present an asymptotic expansion for the height of such trees to constant order term, under the assumption that the sequence of weights behave polynomially. I will then explain the main steps in the proof of this result, which are inspired from those used in the study of the maximal displacement of a branching random walk. Our arguments crucially rely on the description of the ancestry of a vertex (or two vertices) chosen at random in the tree. (Based on arXiv:2101.01156, joint work with Michel Pain.)

# Hao Ni

**The expected signature of the stopped Brownian motion**

*10th May 2021*

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In this talk, we consider the Brownian motion up to the first exit time from a bounded C^{2,\alpha} -domain in R^{d} with 2 <= d <= 8, and prove that the expected signature of such stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development", which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.