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.
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.
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.
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).
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).
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).
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.
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).
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.
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.
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.
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.
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.
13th February 2023
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6th March 2023
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