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Probability seminar abstracts

Our seminars are usually held on Mondays at 1.15pm in room 4W 1.7.

Amanda Turner

Growth of stationary Hasting-Levitov
3rd October 2022
Planar random growth processes occur widely in the physical world. One of the most well-known, yet notoriously difficult, examples is diffusion-limited aggregation (DLA) which models mineral deposition. This process is usually initiated from a cluster containing a single "seed" particle, which successive particles then attach themselves to. However, physicists have also studied DLA seeded on a line segment. One approach to mathematically modelling planar random growth seeded from a single particle is to take the seed particle to be the unit disk and to represent the randomly growing clusters as compositions of conformal mappings of the exterior unit disk. In 1998, Hastings and Levitov proposed a family of models using this approach, which includes a version of DLA. In this talk I will define a stationary version of the Hastings-Levitov model by composing conformal mappings in the upper half-plane. This is proposed as a candidate for off-lattice DLA seeded on the line. We analytically derive various properties of this model and show that they agree with numerical experiments for DLA in the physics literature.

Martin Meier

Forward Induction in a Backward Inductive Manner
10th October 2022
We propose a new rationalizability concept for dynamic games, forward and backward rationalizability, that combines elements from forward and backward induction reasoning. It proceeds by applying the forward induction concept of extensive-form rationalizability in a backward inductive fashion: It first applies extensive-form rationalizability from the last period onwards, subsequently from the penultimate period onwards, keeping the restrictions from the last period, and so on, until we reach the beginning of the game. We show that it is characterized epistemically by (a) first imposing common strong belief in rationality from the last period onwards, then (b) imposing common strong belief in rationality from the penultimate period onwards, keeping the restrictions imposed by (a), and so on. It turns out that in terms of outcomes, the concept is equivalent to the pure forward induction concept of extensive-form rationalizability, but both concepts may differ in terms of strategies. We argue that the new concept provides a more compelling theory for how players react to surprises. In terms of strategies, the new concept provides a refinement of the pure backward induction reasoning as embodied by backward dominance and backwards rationalizability. Finally, it is shown that the concept of forward and backward rationalizability satisfies the principle of supergame monotonicity: If a player learns that the game was actually preceded by some moves he was initially unaware of, then this new information will only refine, but never completely overthrow, his reasoning. Extensive-form rationalizability violates this principle.

Tim Rogers

Structure from Stochasticity
17th October 2022
In this talk I will give an overview of three recent projects connected with the question of how stochasticity can reveal the structure of dynamical systems and even create new structures of its own. There will be no theorems, but there will be lizards, Chebychev polynomials, and people wearing little hats.

David Croydon

Sub-diffusive scaling regimes for one-dimensional Mott variable-range hopping
24th October 2022
I will describe anomalous, sub-diffusive scaling limits for a one-dimensional version of the Mott random walk. The first setting considered nonetheless results in polynomial space-time scaling. In this case, the limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. I will outline how the proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces. The second setting considered concerns a regime that exhibits even more severe blocking (and sub-polynomial scaling). For this, I will describe how, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, I will give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it. This is joint work with Ryoki Fukushima (University of Tsukuba) and Stefan Junk (Tohoku University).

Jess Jay

Second-Class Particles in ASEP under the Blocking Measure
7th November 2022
In this talk we will study the behaviour of second-class particles in the asymmetric simple exclusion process under its natural product blocking measure.

By considering two coupled asymmetric simple exclusion processes we can construct an ASEP with second-class particles. Using the coupling and combinatorial arguments we can find the distribution of the positions of the second-class particles for any given number of second-class particles. In this talk we will discuss this result and see some of the arguments used in the proof.

This is based on joint work with D. Adams and M. Balázs (to feature in an upcoming paper).

Anja Sturm

On the contact process in an evolving edge random environment
14th November 2022
Recently, there has been increasing interest in interacting particle systems on evolving random graphs, respectively in time evolving random environments. In this talk we present some results on the contact process in an evolving edge random environment on (infinite) connected and transitive graphs.

First, we focus on graphs with bounded degree and assume that the evolving random environment is described by an autonomous ergodic spin system with finite range, for example by dynamical percolation. This background process determines which edges are open or closed for infections. Our results concern the dependence of the critical infection rate for survival on the random environment and on the initial configuration of the system. We also consider the phase transition between a trivial/non-trivial upper invariant law and discuss conditions for complete convergence.

Finally, we state some results and open problems in the case of unbounded degree graphs with dynamical percolation as the background such that the contact process evolves on a dynamical long range percolation.

This is joint work with Marco Seiler (University of Frankfurt).

Alex Drewitz

Branching Brownian motion, branching random walks, and the Fisher-KPP equation in spatially random environment
21st November 2022
Branching Brownian motion, branching random walks, and the F-KPP equation have been the subject of intensive research during the last couple of decades. By means of Feynman-Kac and McKean formulas, the understanding of the maximal particles of the former two Markov processes is related to insights into the position of the front of the solution to the F-KPP equation. We will discuss some recent results on extensions of the above models to spatially random branching rates and random nonlinearities.

Apolline Louvet

Some growth properties of the ∞-parent spatial Lambda-Fleming Viot process
31st October 2022
The ∞-parent spatial Lambda-Fleming Viot process (or ∞-parent SLFV) is a measure-valued population genetics process for expanding populations, which is expected to be a space-continuous equivalent of the Eden growth model. Its main feature is the use of "ghost individuals" to fill empty areas and artificially ensure constant local population sizes. In this talk, after defining the process as the unique solution to a martingale problem, we will investigate its growth properties: speed of growth of the front, of the bulk, and scaling of front fluctuations. Our main result is that the front and the bulk both grow linearly in time and at the same speed, which turns out to be much higher than expected due to the reproduction dynamics at the front edge. The proof relies on a (self) duality relation satisfied by the process.

Based on a joint work with Amandine Véber (MAP5, Univ. Paris-Cité) and an ongoing work with Matt Roberts and Jan Lukas Igelbrink (GU Frankfurt and JGU Mainz).

Michel Pain

Extremes of time-inhomogeneous branching Brownian motion
5th December 2022
In this talk, I will discuss a branching Brownian motion model where particles branch into two children at a fixed rate and move according to a Brownian motion with time-dependent variance. I will review the existing literature concerning extremes of this process and present a joint work with Lisa Hartung, Alexandre Legrand and Pascal Maillard where we obtain the convergence of the maximum and of the extremal process when variance is continuously decreasing. In particular, a Bolthausen-Sznitman coalescent appears in the limit.

Tyler Helmuth

The Arboreal Gas
12th December 2022
In Bernoulli bond percolation each edge of a graph is declared open with probability p, and closed otherwise. Typically one asks questions about the geometry of the random subgraph of open edges. The arboreal gas is the probability measure obtained by conditioning on the event that the percolation subgraph is a forest, i.e., contains no cycles. Physically, this is a model for studying the gelation of branched polymers. Mathematically, this is the q=0 limit of the q-random cluster model. I will discuss some of what is known and conjectured about the percolative properties of the arboreal gas.

Based on joint works with R. Bauerschmidt, N. Crawford, and A. Swan.

Yufei Zhang

Exploration-exploitation trade-off for continuous-time reinforcement learning
9th January 2023
Recently, reinforcement learning (RL) has attracted substantial research interests. Much of the attention and success, however, has been for the discrete-time setting. Continuous-time RL, despite its natural analytical connection to stochastic controls, has been largely unexplored and with limited progress. In particular, characterising sample efficiency for continuous-time RL algorithms remains a challenging and open problem.

In this talk, we develop a framework to analyse model-based reinforcement learning in the episodic setting. We then apply it to optimise exploration-exploitation trade-off for linear-convex RL problems, and report sublinear (or even logarithmic) regret bounds for a class of learning algorithms inspired by filtering theory. The approach is probabilistic, involving analysing learning efficiency using concentration inequalities for correlated continuous-time observations, and applying stochastic control theory to quantify the performance gap between applying greedy policies derived from estimated and true models.

Jonathan Jordan

Competing types in preferential attachment graphs with community structure
16th January 2023
We extend the two-type preferential attachment model of Antunović, Mossel and Rácz to networks with community structure. We show that different types of limiting behaviour can be found depending on the choice of community structure and type assignment rule. In particular, we show that, for essentially all type assignment rules where more than one limit has positive probability in the unstructured model, communities may simultaneously converge to different limits if the community connections are sufficiently weak. If only one limit is possible in the unstructured model, this behaviour still occurs for some choices of type assignment rule and community structure. However, we give natural conditions on the assignment rule and, for two communities, on the structure, either of which will imply convergence to this limit, and each of which is essentially best possible. Although in the unstructured two-type model convergence almost surely occurs, we give an example with community structure which almost surely does not converge.

Karen Habermann

A polynomial expansion for Brownian motion and the associated fluctuation process
6th February 2023
We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with a general study of the asymptotic error arising when approximating the Green's function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green's function of a regular Sturm-Liouville problem and for the Green's function associated with the classical orthogonal polynomials.

Félix Foutel-Rodier

The genealogy of a nearly critical branching process in varying environment
13th February 2023
Branching processes in varying environment (BPVEs) are a natural extension of Galton-Watson processes, where the distribution of the number of children depends on the generation. In this work, we define a notion of near criticality for BPVEs. We show that the genealogy of a near critical BPVE, viewed as a random metric space, converges to a limiting tree in the Gromov-Hausdorff-Prohorov topology. This limit is expressed as a time-change of the corresponding limit for Galton-Watson processes: the Brownian coalescent point process. This work also illustrates and extends a general approach to tackle convergence of genealogies for branching processes which I will discuss. It relies on computing the so-called moments of the genealogy using a many-to-few formula.

Peter Mörters

The contact process on dynamical scale-free networks
16th February 2022
Note unusual day and room: 6W 1.2
We investigate the contact process on two different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent vertices. Depending on the type of graph, the tail exponent of the degree distribution and the updating rate, we find parameter regimes of fast and slow extinction and in the latter case identify metastable exponents that undergo first order phase transitions. Joint work with Emmanuel Jacob (ENS Lyon) and Amitai Linker (Universidad Andrés Bello).

Noah Halberstam

Infinite trees in the arboreal gas
20th February 2023
The arboreal gas, alternatively known as the edge weighted unrooted spanning forest model, is equivalent to Bernoulli percolation conditioned to be acyclic. Recent exciting work of Bauerschmidt, Crawford and Helmuth has established in dimensions d>2 the existence of a value for the edge weight parameter above which certain infinite volume limits contain an infinite tree almost surely, and thus that the model demonstrates a phase transition with respect to this parameter. Using probabilistic techniques, we show that in low dimensions d=3,4 the infinite tree is unique, and give strong heuristic evidence that the number of infinite trees is in fact infinite in higher dimensions. We also prove that in any dimension all such infinite trees must be one-ended almost surely. Joint work with Tom Hutchcroft.

Nic Küpper

The largest component in a subcritical Soft Random Geometric Graph
27th February 2023
Random Geometric Graphs have long been studied in the context of percolation as continuum analogues of lattice models. We can create a mixed model by first sampling a normal Gilbert graph, that is we sample a Poisson point process with intensity λ and connect any two points that are within distance 1 of each other. We then, independently of everything else, delete edges with probability 1-p, i.e. we run percolation with parameter p on the edges of the Gilbert graph. Our goal is to understand the largest component when restricting this resulting graph to some box. We show that the size of the largest component scales with the logarithm of the size of the box.

Dominic Yeo

The critical window for random transposition random walk
6th March 2023
The random walk on the permutations of [N] generated by the transpositions was shown by Diaconis and Shahshahani to mix with sharp cutoff around N log N /2 steps. However, Schramm showed that the distribution of the rescaled relative lengths of the largest cycles converge considerably earlier, after (1+ε)N/2 steps. We show that this behaviour emerges precisely during the same critical window as for the Erdos-Renyi random graph process. Our methods are rather different, and include metric scaling limits and the structure of directed cycles within large 3-regular graphs. Ongoing joint work with Christina Goldschmidt.

Angelica Pachon

Upper bounds for the largest component in critical inhomogeneous random graphs
13th March 2023
Numerous random graphs inspired in real networks are inhomogeneous in the sense that not all vertices have the same characteristics which may influence the connection probabilities between pairs of vertices. In this talk, I will start by presenting the most known inhomogeneous random graph models. Next, I will consider the Norrous-Reittu random graph model where edges are present independently, but edge probabilities are moderated by vertex weights. On the critical regime this model is studied by van der Hofstad (2012) using a branching process approximation.

Finally, I will present the results of a joint work with Umberto De Ambroggio where we analyse the order of the maximal component of this model, in the critical regimen. We use probabilistic arguments based on martingales and adapt some ideas originally introduced by Nachmias and Peres (2010).

Brett Kolesnik

Random tournaments
20th March 2023
A tournament on a graph is an orientation of its edges. Vertices are players and each edge is a game, directed toward the winner. In this talk, we will present some recent results on random tournaments. With David Aldous (Berkeley) we construct random tournaments using Strassen’s coupling theorem, yielding a probabilistic proof of Moon's classical theorem. With Mario Sanchez (Cornell) we study the geometry of random tournaments, with its connections to permutahedra, zonotopes, etc. We show that the recent Coxeter permutahedra are related to tournaments that involve collaboration (and competition, as usual) answering a question of Stanley. Finally, we settle a conjecture of Takács about the asymptotic number of score sequences. The proof involves combinatorics (Erdős–Ginzburg–Ziv numbers), renewal theory and infinitely divisible probability distributions.

Pawel Rudnicki

Reinforced digging random walks with linear reinforcement
27th March 2023
The branching-ruin number of a tree was shown to be the critical parameter for recurrence and transience of the once-reinforced random walks. Later, the same was proven for the random walk with random conductances and the M-digging random walk. In the talk, we will further explore the relation between the branching-ruin number and the criticality of random processes on trees. We will introduce reinforced digging random walks (RDRW), which are self-interacting non-Markovian random walks that depend on the reinforcement parameter d > 0. In particular, we will study RDRW with a linear reinforcement and show that the phase transition for recurrence and transience depends on the relation between the branching-ruin number of a tree and a certain quantity K(d) that will be explicitly defined in the talk.

Omer Bobrowski

Homological Connectivity in Random Geometric Complexes and Poisson Approximation
17th April 2023
A well-known phenomenon in random graphs is the phase-transition for connectivity, proved first by Erdős and Rényi in 1959. In this talk we will discuss a high-dimensional analogue of this phenomenon known as "homological connectivity". Loosely speaking, homology is an algebraic-topological structure describing various types of "cycles" that can be formed in high-dimensional shapes. Considering an increasing sequence of shapes, homological cycles a formed and filled in at various times. Homological connectivity is the point where the homology of such sequences stops changing, or "stabilizes". The model we study is the random Čech complex, which is a high-dimensional generalisation of the random geometric graph. We will show that there is a sequence of sharp phase transitions (for different degrees of homology). In addition, we will show that in each critical window, the obstructions to homological connectivity have a functional Poisson process limit.

Elnur Emrah

Classifying boundary fluctuations for uniformly random Gelfand-Tsetlin patterns
24th April 2023
A Gelfand-Tsetlin (GT) pattern of depth n is an interlacing array of n(n+1)/2 real entries distributed over n levels such that level k in {0, 1, ..., n-1} contains exactly n-k entries. This object naturally arises from an n by n Hermitian matrix by placing the eigenvalues of the (n-k) by (n-k) leading principal submatrix on level k. Let G(n) denote a GT pattern of depth n with a fixed increasing sequence a(n) in Rn on level 0 and with the remaining entries viewed as particles chosen uniformly at random. In this talk, our interest is in the limiting boundary fluctuations of G(n) at the level of finite-dimensional distributions of first particles. We present a classification theorem that identifies five fluctuation regimes in terms of the level zero data {a(n): n \in Z_{>0}} and describe the corresponding limit processes. This result is from a forthcoming joint work with Kurt Johansson.

Adam Bowditch

Anomalous fluctuations of random walks
9th May 2023 - note unusual day and room - Tuesday in 6W 1.2
Understanding the interplay between the geometry of a random space and the stochastic processes that live upon it has been a major focus of probability over the last four decades. In this talk we will focus on the central example of a random walk on a supercritical percolation cluster on the integer lattice. The inhomogeneity of the environment incorporates traps from which the walk will take an anomalously long time to escape. Our main focus will be to discuss how the trapping is exacerbated by adding a bias and understand the different asymptotic behaviours that can arise.

Sonny Medina Jimenez

Excursions from hyperplanes for the α-stable process
15th May 2023
Breaking the paths of isotropic Lévy processes in radial excursions is a technique that has been used to understand the behaviour of processes when they approach the origin or the unit sphere. We aim to build on these ideas and techniques and consider the behaviour of excursions from minima/maxima (in the direction of a given vector) for the α-stable Lévy process in dimension d≥2. We will use the associated path decomposition to address issues such as the characterisation of ladder processes, fluctuations in first passage times, and the so-called deep Wiener-Höpf factorisation. We will also show how to construct conditional laws of the process with respect to hyperplanes and half-spaces.

Matthieu Jonckheere

The critical parameter of Fermat distance
22nd May 2023
We first review some classical methods in machine learning to deal with dimension reduction and distance learning. We then elaborate on a new density-based estimator for weighted geodesic distances that takes into account the underlying density of the data, and that is suitable for nonuniform data lying on a manifold of lower dimension than the ambient space. The consistency of the estimator is proven using tools from first passage percolation. The macroscopic distance obtained depends on a unique parameter and we discuss the choice of this parameter and the properties of the obtained distance for machine learning tasks.