Math Probability Mini-Course
Tuesdays, November 8-December 6, 2016
Obtaining a rigorous mathematical description of spatial growth phenomena, such as the growth of a colony of bacteria, or lightning formation during a thunderstorm, is an interesting and challenging problem, which has motivated the introduction of several probabilistic models in recent years. In this course we will focus on random aggregation, describing growth as the result of subsequent aggregation of particles. Many mathematical models of random aggregation are available in the literature, often assuming that particles move on a discrete structure, e.g. the d-dimensional lattice Zd, or an in infinite graph. Examples of such models are Diffusion Limited Aggregation (DLA in short), Internal DLA, the Eden model, Dielectric Breakdown models and many more. While understanding the behaviour of these models is a very interesting mathematical challenge, some of their features have been empirically found to depend on the choice of the underlying graph, which is somewhat unsatisfactory. This led researchers to look for o {lattice alternatives, so as to completely bypass the choice of the discretization. In this short course I will discuss one instance of the so called Hastings-Levitov models, introduced in 1998 by Hastings and Levitov [HL98] following ideas of Carleson and Makarov [CM01], who first proposed to use complex analysis techniques to build clusters of particles in the complex plane.
Hastings-Levitov models encode the growth of an increasing family of compact sets K0 ⊂ K1 ⊂ K2 ⊂ … in the complex plane, called clusters, built by iterated composition of conformal maps. At each step, a new particle, consisting of a (sufficiently nice) compact set, is attached to the current cluster at a location chosen according to the harmonic measure of the cluster boundary seen from ∞, as in discrete DLA. We think of particles as being small, while their number diverges to infinity. Crucially, the size of the nth particle depends on the geometry of the cluster Kn-1 close to the attachment point, hence being itself random. When no correction is introduced to account for the randomness in the size of particles, the cluster turns out to be the result of iterated composition of independent random conformal maps. Such model is referred to as HL (0), and it will be the main object of study in this course. One could, on the other hand, trade independence of conformal maps for some control on the particle size. This leads to the definition of a one parameter family of models called HL (α), the parameter α∈R≥0 measuring the strength of such control. Although we will not discuss the case α > 0 in the course, I will briefly comment on why it is interesting and what additional challenges it presents with respect to the α= 0 case.
Specializing to the case α = 0, a natural question to ask is how does an HL (0) cluster look like after a large number of arrivals. In other words, if we let the number of particles diverge to infinity, while sending the size of each particle to zero, can we identify a limiting shape for HL (0) clusters? The answer turns out to be positive: in 2012 Norris and Turner proved that large HL (0) clusters converge to in inflating discs in the small particle limit, independently of the specific particle shape. Moreover, they obtained a very precise description of the internal structure of the clusters, which they showed to converge to the Brownian Web. In the first part of the course I will discuss some heuristic arguments in support of the scaling limit result, to then sketch the proof of it presented in [NT12]. Time permitting, I will also comment on the Brownian Web result.
Once the sequence of random conformal maps encoding the cluster growth has been shown to converge to a deterministic limit, the question arises of a description of the fluctuations around it. In the second part of the course I will review some recent results in this direction [Sil15]. More precisely, I will show that these fluctuations are described, in logarithmic coordinates, by a random Gaussian holomorphic function F, of which I will provide an explicit construction. In doing so, I will identify the boundary values of F, which describe, in a suitably weak sense, the fluctuations of the actual cluster boundary. I will sketch the proofs of these results, hoping to highlight the main ideas while leaving the reader to fill in the details. Finally, I will briefly explain how this fits into a global description of the clusters' fluctuations in time, where the continuous time parameter controls the asymptotic diameter of the clusters in the small particle limit. We will see that the boundary values of F perform an Ornstein-Uhlenbeck process on the space of tempered distributions, which converges, as the diameter of the clusters is left to grow indefinitely, to a well-known log-correlated Gaussian field on the unit circle. Although I will not present a proof of this fact, I will discuss a heuristic argument explaining the appearance of such OU process. Finally, I will conclude by presenting some open problems on Hastings-Levitov growth and related models.
References
- [CM01] Lennart Carleson and Nikolai Makarov. Aggregation in the plane and Loewner's equation. Communications in Mathematical Physics, 216(3):583{607, 2001.
- [HL98] Matthew B. Hastings and Leonid S. Levitov. Laplacian growth as one-dimensional turbulence. Physica D: Nonlinear Phenomena, 116(1):244{252, 1998.
- [NT12] James Norris and Amanda Turner. Hastings-Levitov aggregation in the small-particle limit. Comm. Math. Phys., 316(3):809{841, 2012.
- [Sil15] Vittoria Silvestri. Fluctuation results for Hastings-Levitov planar growth. Probability Theory and Related Fields, pages 1{44, 2015.
Biography
Vittoria Silvestri is a Research Fellow at Jesus College, and member of the Probability group at the University of Cambridge. She has just completed her Ph.D. in Mathematics within the Cambridge Centre for Analysis (CCA). Before going to Cambridge, she studied Mathematics at the University of Rome La Sapienza, Italy, where she obtained her B.Sc. and M.Sc. degrees. She is mainly interested in Stochastic Processes and their applications to real-world problems.