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## Geometry and Analysis of Random Processes## UK Probability Easter Meeting 2013## 8 - 12 April 2013, Centre for Mathematical Sciences |
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Certain natural measures on half planar random maps enjoy what we call (in analogy to SLE) a "domain Markov" property. I will explore this property, describing work with Gourab Ray, as well as an application of this property to study exponents for critical percolation on several models of random planar maps (work with Nicolas Curien).

The inclusion process is an interacting particle system where particles perform independent random walks with a diffusion rate d in addition to an 'inclusion' effect. The rates for inclusion jumps are proportional to the product of the occupation numbers on departure and target site. In the limit of vanishing diffusion rate acondensation phenomenon occurs where all particles concentrate on a single site in a typical stationary configuration. We focus on the totally asymmetric one-dimensional case with nearest neighbour jumps. Our aim is to analyze the dynamics of the condensate's emergence in the thermodynamic limit with fixed average particle density. The whole time evolution can be divided into four regimes, nucleation, coarsening, saturation and stationarity. We describe each of them heuristically, with a particular emphasis on the power law behaviour in the coarsening regime.

*This is a joint work with Paul Chleboun and Stefan Grosskinsky.*

The basic challenge of mathematical population genetics is to understand the relative importance of the different forces of evolution in shaping the genetic diversity that we see in the world around us. This is a problem that has been around for a century, and a great deal is known. However, a proper understanding of the role of a population's spatial structure is missing. Recently we introduced a new framework for modelling populations that evolve in a spatial continuum. In this talk we briefly describe this framework before outlining some preliminary results on the importance of spatial structure for natural selection.

Consider the complete graph on n vertices with independent and identically distributed edge-weights having some absolutely continuous distribution. The minimum spanning tree (MST) is simply the spanning subtree of smallest weight. It is straightforward to construct the MST using one of several natural algorithms. Kruskal's algorithm builds the tree edge by edge starting from the globally lowest-weight edge and then adding other edges one by one in increasing order of

weight, as long as they do not create any cycles. At each step of this process, the algorithm has generated a forest, which becomes connected on the final step. In this talk, I will explain how its possible to exploit a connection between the forest generated by Kruskal's algorithm and the Erdös-Rényi random graph in order to prove that M_n, the MST of the complete graph, possesses a scaling limit as n tends to infinity. In particular, if we think of M_n as a metric space (using

the graph distance), rescale edge-lengths by n^{-1/3}, and endow the vertices with the uniform measure, then M_n converges in distribution in the sense of the Gromov-Hausdorff-Prokhorov distance to a certain random measured real tree.

This is joint work with Louigi Addario-Berry (McGill), Nicolas Broutin (INRIA Paris-Rocquencourt) and Grégory Miermont (ENS Lyon).

Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed.

I will give an overview of a new theory of "regularity structures" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of "renormalisation group techniques" in this context.

At the technical level, the main novel idea involves a complete rethinking of the notion of "Taylor expansion" at a point for a function or even a distribution. The resulting structure is useful for encoding "recipes" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.

Commencing with joint work with David Aldous, I have been investigating the use of Poisson line processes in solving frustrated optimization problems for transportation networks. I will survey this, and then discuss work in progress concerning a curious random metric space on the plane which can be constructed with the help of an *improper* Poisson line process.

We look at a general two-sided jumping strictly alpha-stable process where alpha is in (0,2). By censoring its path each time it enters the negative half line we show that the resulting process is a positive self-similar Markov Process. Using Lamperti's transformation we uncover an underlying driving Lévy process and, moreover, we are able to describe in surprisingly explicit detail the Wiener-Hopf factorization of the latter. Using this Wiener-Hopf factorization together with a series of spatial path transformations, it is now possible to produce an explicit formula for the law of the original stable processes as it first ``enters'' a finite interval, thereby generalizing a result of Blumenthal, Getoor and Ray for symmetric stable processes from 1961.

This is joint work with Alex Watson (Bath) and JC Pardo (CIMAT)

This would be based on: arxiv.org/abs/1303.6894, but minus details and plus some pictures

Although self-avoiding walk is very easy to define, surprisingly little is known about it. In this talk, we will present two results on the delocalization of the endpoint of a uniform self avoiding walk of given length on the d-dimensional square lattice for d ≥ 2.

We show that the probability for a walk of length n to end at a point x tends to 0 as n tends to inﬁnity, uniformly in x. Also, for any ﬁxed x, we prove a quantitative bound for the above probability. n particular, this provides a bound on the probability that a self-avoiding walk is a polygon.

Joint work with Hugo Duminil-Copin, Alexander Glazman and Alan Hammond.

The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been an exciting area in probability theory over the last decade because Schramm’s curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion, and how this perspective can be used to study the sample path behavior of SLE. This talk is based on joint work with Scott Sheffield.

We introduce and study a model of frozen percolation on a graph where edges open at constant rate 1 and clusters freeze at a rate alpha, independently of their size. This is straightforward to define on a finite graph, and we show that an infinite volume limit exists when the graph is amenable and vertex-transitive. We will then talk about the percolative properties of the final configuration as a function of alpha. We believe that the model exhibits a phase transition, and that infinite clusters appear when alpha is less than some critical value. Depending on the dimension, some arguments suggest that more than one infinite clusters might coexist. Moreover, the system shows signs of self-organised criticality.

We will investigate the geometry of large unicellular maps in genus whose size is linear in the number of vertices. Some recently obtained results and conjectures will be discussed.

In an Erdos-Renyi random graph above the phase transition, i.e., where there is a giant component, the size of (number of vertices in) this giant component is asymptotically normally distributed, in that its centred and scaled size converges to a normal distribution. This statement does not tell us much about the probability of the giant component having exactly a certain size. In joint work with Bela Bollobas we prove a `local limit theorem' answering this question for hypergraphs; the graph case was settled by Luczak and Luczak. The proof is based on a `smoothing' technique, deducing the local limit result from the (much easier) `global' central limit theorem.

A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on the integer points in [0,n) without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions.

Joint work with Y. Babichenko, Y. Peres, R. Peretz and P. Winkler.

We prove a CLT under $\sqrt{t \log t}$ scaling for the displacement of the (non-Poissonian) random flight process obtained (in earlier work

of Marklof and Strombergsson) in the Boltzmann-Grad limit applied to the periodic Lorentz gas in 2d.

(Joint work in progress with Jens Marklof)

I will describe how branching random walks (with gaussian increments) and comparison theorems can be used to deduce the convergence of he law of recentered maxima of the discrete, two dimensional Gaussian free field.

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