Probability

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.

Universality in disordered systems has always played a central role inthe direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class. In this talk,  I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain, to the top edge in a game of Tetris. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class. Here, I will talk about some recent results on universal geometric properties of the KPZ fixed point and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models.

The talk is based on joint works with Jeremy Quastel and Balint Virag.

In this talk, we present recent results on an anisotropic variant of the Kardar-Parisi-Zhang equation, the Anisotropic KPZ equation (AKPZ), in the critical spatial dimension d=2. This is a singular SPDE which is conjectured to capture the behaviour of the fluctuations of a large family of random surface growth phenomena but whose analysis falls outside of the scope not only of classical stochastic calculus but also of the theory of Regularity Structures and paracontrolled calculus. We first consider a regularised version of the AKPZ equation which preserves the invariant measure and prove that, at large scales, the correlation length grows like t1/2 (log t)1/4 up to lower order correction. Second, we prove that in the so-called weak coupling regime, i.e. the equation regularised at scale N and the coefficient of the nonlinearity tuned down by a factor (log N)-1/2, the AKPZ equation converges to a linear stochastic heat equation with renormalised coefficients. Time allowing, we will explain how the tools and techniques above can be used to tackle a variety of other systems which were before out of reach, e.g. (super-)critical SPDEs and diffusions in divergence free ergodic vector fields. This is joint work with D. Erhard, M. Gubinelli and F. Toninelli.

We shall discuss the quantum dynamics associated with ergodic Schroedinger operators with singular continuous spectrum. Upper bounds on the transport moments have been obtained for several classes of one-dimensional operators, particularly, by Damanik--Tcheremchantsev, Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method which allows to recover most of the previous results and also to obtain new results in one and higher dimensions. The input required to apply the method is a large-deviation estimate on the Green function at a single energy. Based on joint work with S. Sodin.

We consider a variant of the configuration model with an embedded community structure, where every vertex has an internal and an outgoing number of half edges. We pick a uniform matching of the half edges subject to the constraints that internal edges in each community are matched to each other and the proportion of half edges between communities i and j being Q(i,j). We prove that a simple random walk on the resulting graph G=(V,E) exhibits cutoff if and only if the product of the Cheeger constant of Q times log|V| diverges.  (Joint work with Jonathan Hermon and Perla Sousi.)

We consider a fully-connected loss network with dynamic alternative routing, each link of capacity K. Calls arrive to each link {i, j} at rate λ independently and depart at rate 1. If the link is full upon arrival, a third node k is chosen uniform and the call is routed via k: it uses a unit of capacity on both {i, k} and {k, j} if both have spare capacity; otherwise, the call is lost. This is a model for telephone networks, implemented by BT in the 1990s. We analyse the asymptotics of the mixing time of this process, depending on the traffic intensity α := λ/K. In particular, we determine a phase transition at an explicit threshold α*: there is fast mixing if α < α* or α > 1, but metastability if α* < α < 1. We also discuss a fixed for metastability — ie, an adjustment to the model which removes the slow-mixing phase. Again, this was implemented by BT in the UK telephone network.

The Widom-Rowlinson model is one of the few continuum models of classical statistical mechanics for which the existence of a phase transition has been rigorously proven. In the single-color version the energy is roughly the area covered by a union of balls. We investigate the model in a finite 2-dimensional box in the limit of large density and ask (1) what is the probability that the union of disks centered at particles is approximately equal to a given shape, say a disk, and (2) how should we think of interface fluctuations in the vicinity of the disk. The questions are motivated by the conjectured metastable behavior for a dynamic model in which particles are born and die. The talk presents partial answers as well as connections with geometric inequalities and the parabolic hull and growth processes introduced for studying the convex hull of a random set of points. Based on joint work with Frank den Hollander, Roman Kotecký and Elena Pulvirenti.

We consider facilitated exclusion processes (FEP) in one dimension. These models belong to a class of kinetically constrained lattice gases, the process was introduced in the physics literature motivated by studying the active-absorbing phase transition. Under the dynamics a particle can move to a neighbouring site provided that the target site is empty (the exclusion rule) and the other neighbour of the departure site is occupied (the constraint). These processes have recently attracted a lot of attention due to their interesting hydrodynamic limit behaviour. We examine the mixing time, the time to reach equilibrium, on an interval with closed boundaries and also with periodic boundary conditions. On the interval we observe that asymmetry significantly changes the mixing behaviour. The analysis naturally splits into examining the time to reach the ergodic configurations (irreducible component) followed by the time needed to mix on this set of configurations. This is joint work with James Ayre (Oxford).