## Seminars 2015/16

## Prob-L@B Internal Monthly Seminar

Our internal seminar is usually held on the last Thursday of the month, at 12.15pm in 4W1.7. For more information please contact Cécile Mailler or see the PIMS website.

29.10.15 **Bati Sengul - **Cutoff for conjugacy-invariant random walks on the permutation group

26.11.15 **Antal Jarai - **Inequalities for critical exponents in d-dimensional sandpiles

17.12.15 **Alex Cox -** (Martingale) optimal transport, the Skorokhod embedding problem and measure valued martingales**
**25.02.16

**Matt Roberts -**Robustness of mixing and bottleneck sequences

**31.03.16**

**Simon Harris**- Branching Brownian motion and the Van Saarloos expansion

**28.04.16**

**Alexandre Stauffer**- Multi-particle diffusion limited aggregation

**26.05.16**

**Peter M**

**örters -**The contact process on evolving scale-free networks

## Seminar abstracts

### 5th October 2015 - Will Perkins (University of Birmingham)

**An excluded volume approach to hard spheres, independent sets, and matchings**

What is the probability that a random geometric graph contains no edges? This quantity is in fact the partition function of the hard sphere model from statistical physics. I will present a new technique for bounding this partition function using estimates on the expected excluded volume. I will then show how related ideas can give tight bounds on the occupancy fraction in two discrete models with hard constraints: the hard core lattice gas model and the monomer-dimer model.

### 12th October 2015 - Alex Hening (University of Oxford)

**The free path in a high velocity random flight process associated to a Lorentz gas in an external field**

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.

### 19th October 2015 - Lutz Warnke (University of Cambridge)

**The phase transition in bounded-size Achlioptas processes**

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges. Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph.

We shall prove that, for a large class of widely studied rules (so-called bounded-size rules), the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process. For example, assuming \eps^3 n \to \infty and \eps \to 0 as n \to \infty, the size of the largest component after step t_c n \pm \eps n whp satisfies L_1(t_c n-\eps n) \sim D \eps^{-2}\log(\eps^3 n) and L_1(t_c n+\eps n) \sim d\eps n, where t_c,D,d>0 are rule-dependent constants (in the Erdös-Rényi process we have t_c=D=1/2 and d=4).

Based on joint work with Oliver Riordan.

### 26th October 2015 - Oriane Blondel (Université Lyon 1)

**Random walk on environments with spectral gap**

We consider random walks in random environments with positive spectral gap. Under suitable assumptions on the "asymmetry" of the jump rates w.r.t. the reference measure, we identify the invariant measure for the process seen from the walker and establish a law of large numbers and invariance principle for the walker. These findings rely on a general result concerning L^2 perturbations of Markov generators. We have in mind applications to random walks on a toy model for glassy systems.

Joint work with Luca Avena and Alessandra Faggionato.

### 2nd November 2015 - Vittoria Silvestri (University of Cambridge)

**Hastings-Levitov growth and the GFF**

I will discuss one instance of the so called Hastings-Levitov planar aggregation model, consisting of growing random clusters on the complex plane, built by iterated composition of random conformal maps. In 2012 Norris and Turner proved that in the case of i.i.d. maps the limiting shape of these clusters is a disc. In this talk I will show that the fluctuations around this asymptotic behaviour are given by a random holomorphic Gaussian field F on {||z||>1}, of which I will provide an explicit construction. The boundary values of F perform a Gaussian Markov process on the space of distributions, which is conveniently described as the solution of a stochastic PDE. When the cluster is allowed to grow indefinitely, this process converges to the restriction of the whole plane Gaussian Free Field to the unit circle.

### 9th November 2015 - Rob Jack (University of Bath)

**Large deviations and effective interactions in physical systems**

We consider time-averaged measurements in physical systems described by stochastic dynamics. In typical cases of interest, these measurements obey a large-deviation principle with a speed proportional to the averaging time. (This result follows from work by Donsker and Varadhan in the 1970s.) I will review some recent work on these kinds of large deviation in physics, some of which is summarised in [1]. In particular, there are several cases where the sample paths associated with the large deviation events exhibit interesting structure, and differ qualitatively from typical sample paths. I will discuss the characterisation of this rare structure via the construction of auxiliary systems for which it becomes typical.

[1] R. L. Jack and P. Sollich, Eur. Phys. J.: Special Topics 224, 2351 (2015). doi:10.1140/epjst/e2015-02416-9

### 16th November 2015 - Lorenzo Zambotti (Université Paris VI)

**The renormalisation group in regularity structures**

We discuss a new approach to the renormalisation group in the theory of regularity structures, based on a Hopf algebra and a Hopf module of finite labelled forests which represent suitable integral kernels. I will try to explain this algebraic-combinatorial construction in the simplest possible terms without assuming any knowledge of regularity structures or stochastic PDEs. (Based on joint work with Yvain Bruned and Martin Hairer.)

### 23rd November 2015 - Vladislav Vysotsky (Arizona State/Imperial College/St. Petersburg Division of Steklov Institute)

**The largest gap problem for random walks**

We aim to describe how sparse is the set of the first n values of a one-dimensional random walk. We assume that the increments of the walk have a zero mean and finite variance. Our main result is a limit theorem for the size of the largest gap (maximal spacing) within the range of the walk by the time n. In addition, for integer-valued walks we obtain a similar statement on the number of non-visited sited within the convex hull of the range. The proofs are based on our results on the tail asymptotic for the hitting times of bounded intervals by random walks.

### 30th November 2015 - Stephen Muirhead (University College London)

**Quenched localisation in slowly varying trap models**

Trap models with slowly varying trap distributions constitute one of the three basic regimes of trap models (along with integrable and stable traps), and arise naturally in the study of certain random walks in random media, such as biased random walks on critical structures, spin-glass dynamics on sub-exponential time scales, and recurrent random walks in random environments. In this talk I will discuss recent progress (joint with David Croydon) on understanding quenched localisation properties of slowly varying trap models, which turn out to be surprisingly delicate. Our main result concerns a simple effective trap model -- the Bouchaud trap model on the positive integers -- for which we demonstrate that there exist slowly varying trap distributions such that quenched localisation occurs on exactly N sites, for any integer N greater or equal to two. A key ingredient is an observation about the almost sure limit superior of the sum/max ratio of i.i.d. sequences of slowly varying random variables, which appears to be new.

### 7th December 2015 - Thomas Mountford (EPFL Lausanne)

**Stability of multiserver queuing systems**

We consider a system where jobs arrive as a "Poisson Rain" onto an infinite lattice of servers. Each job has a random service time and a random collection of servers required to perform the job. We give a good criterion for stability of the system (at sufficiently low rates of arrival) and discuss uniqeness of the equilibrium in the stability regime.

Joint with Sergey Foss and Takis Konstantopoulous.

### 6th January 2016 - Kevin Durant (Stellenbosch University)

**Betweenness Centrality in Random Trees**

Betweenness centrality is a quantity that is frequently used in graph theory to measure how "central" a vertex v is. It is defined as the sum, over pairs of vertices other than v, of the proportions of shortest paths that pass through v. Here, we study the distribution of betweenness centrality in random trees. Specifically, we show that the betweenness centrality of a randomly chosen vertex in a random tree of size n is usually of linear order, whereas the average over all vertices is of order n^{3/2}, and the maximum is of order n^2. We also obtain limiting distributions for the betweenness centrality of the root vertex and for the maximum betweenness centrality; the latter by means of a limiting structure known as the continuum random tree. Finally, we discuss the related class of subcritical graphs, as well as random recursive trees.

### 11th January 2016 - Matt Aldridge (University of Bath)

**Recent results in group testing**

Group testing is the following problem: There are many items, some of which are “defective”. One can test subsets of items to discover whether each subset contains zero defective items or at least one defective item. How many such tests are required to accurately discover which items are defective? This problem has applications in biostatistics, genomics, communications, and elsewhere.

In this talk, we discuss some recent results in group testing. We concentrate on “nonadaptive” testing, when all the tests have to be designed beforehand and cannot be altered given earlier test results, and in particular Bernoulli group testing, where each item is placed in each test at random, independently with some fixed probability. We look at limits on the number of tests required, and practical algorithms for achieving performance close to these limits.

Joint work with Oliver Johnson and Leonardo Baldassini.

### 1st February 2016 - Erwin Bolthausen (University of Zurich)

**On the Thouless Anderson Palmer equations for the perceptron in spin glasses**

The perceptron is a simple neural network. In connection with determining the memory capacity of the net, there arose a number of mathematically interesting spin glass problems. The model which appears in this context is considerably more delicate than the well-known Sherrington-Kirkpatrick model. In particular, a complete Parisi-picture is presently mathematically out of reach. We discuss the so-called TAP equations (after Thouless, Anderson, Palmer), and their connection with Talagrand’s formula for the (high temperature) free energy.

### 8th February 2016 - Codina Cotar (University College London)

**Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs**

We introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge- and to vertex-reinforced random walks with super-linear reinforcement on arbitrary in infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs.

For edge-reinforced random walks, we complete the results of Limic and Tarres [Ann. Probab. (2007)] and we settle a conjecture of Sellke [Technical Report 94-26, Purdue University (1994)] by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times.

For vertex-reinforced random walks, we extend results previously obtained on Z by Volkov [Ann. Probab. (2001)] and by Basdevant, Schapira and Singh [Ann. Probab. (2014)], and on complete graphs by Benaim, Raimond and Schapira [ALEA (2013)]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times.

(This is joint work with Debleena Thacker.)

### 15th February 2016 - Richard Pymar (University College London)

**Mixing time of the exclusion process on hypergraphs**

We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant C such that for any regular uniform connected hypergraph G, the epsilon-mixing time of the exclusion process on G with any feasible number of particles can be upper-bounded by C T_{EX(2,G)} log(|V|/epsilon), where |V| is the number of vertices in G and T_{EX(2,G)} is the 1/4-mixing time of the corresponding exclusion process with 2 particles. The proof involves an adaptation of techniques invented by Morris and developed by Oliveira. This is (ongoing) joint work with Stephen Connor.

### 22nd February 2016 - Horatio Boedihardjo (University of Reading)

**A pathwise estimate on iterated integrals**

Rough path theory has recently emerged as a new way of looking at stochastic differential equation. It allows us to do calculus with non-semi-martingales and strengthens results on large deviation principles and stochastic flows. The theory gives naturally Stratonovich integration. To cover Ito-type of calculus, Gubinelli recently proposes a new theory of rough path and left open a basic conjecture on the decay rate of iterated integrals. I will discuss an answer to this conjecture.

### 29th February 2016 - Jan Swart (UTIA, Prague)

**Rank-based Markov chains, self-organized criticality, and order book dynamics**

In this talk, we will take a look at some systems of interacting particles on the real line, where the only spatial structure that is relevant for the dynamics is the relative order of the particles. Examples of such systems are the modified Bak-Sneppen model, introduced (as a variation of the original 1993 model) by Meester and Sarkar (2012), Barabasi's (2005) queueing system and a variation on the latter due to Gabrielli and Caldarelli (2009), a model for the evolution of the state of an order book on a stock market, introduced by Stigler (1964) and independently by Luckock (2003), and a two models for canyon formation introduced by me (2014). All these systems employ a version of the rule "kill the lowest particle" and seem to exhibit self-organized criticality at a critical point that marks the boundary between an interval where all particles are eventually removed and an interval where particle stay in the system forever.

### 7th March 2016 - Vladimir Vovk (Royal Holloway)

**Selected results in probability-free continuous-time finance**

This talk will describe an approach to mathematical finance whose main advantage over the usual probabilistic approaches is that it does not presuppose a given probability measure and its results depend on economic considerations combined only with topological (but not statistical) assumptions. The price to pay is the relative paucity of such results. Still, under the continuity assumption, it is possible to show that a typical continuous price path "looks like Brownian motion" with a possibly deformed time axis, and the weaker assumption of boundedness of jumps implies the existence of pathwise stochastic integrals of functions with finite p-variation for some p with respect to typical cadlag price paths with bounded jumps.

### 14th March 2016 - Lorna Wilson (University of Bath)

**A spectrum of regularity: the impact of periodicity on the zero-crossings of random functions**

This talk will cover my research on the effect of an oscillatory auto-correlation function on the zero-crossings of a Gaussian process. I will also talk briefly about my role as a commercial research assistant at the University of Bath.

Continuous random processes are used to model a huge variety of real world phenomena. In particular, the zero-crossings of such processes find application in modelling processes of diffusion, meteorology, genetics, finance and applied probability. To identify the Probability Density Function (PDF) for the times between successive zero-crossings of a stochastic process is a challenging problem with a rich history. I will talk about the methods used to investigate the inter-event PDF for a Gaussian process with a specific, oscillatory auto-correlation function. I found that increasing the rate of oscillations in the auto-correlation function results in more `deterministic' realisations of the process. In an ever-more complex world, the potential applications for this scale of `regularity' in a random process are far reaching and powerful.

### 4th April 2016 - Tom Hutchcroft (University of British Columbia)

**Update tolerance in uniform spanning forests**

The uniform spanning forests (USFs) of an infinite graph G are defined to be infinite volume limits of uniformly chosen spanning trees of finite subgraphs of G. These limits can be taken with respect to two extremal boundary conditions, yielding the free uniform spanning forest (FUSF) and wired uniform spanning forest (WUSF). While the wired uniform spanning forest has been quite well understood since the seminal paper of Benjamini, Lyons, Peres and Schramm ('01), the FUSF is less understood, and some very basic questions about it remain open. In this talk I will introduce a new tool in the study of USFs, called update tolerance, and describe how update tolerance can be used to prove, among other things, that the FUSF has either one or infinitely many connected components on any infinite Cayley graph, and that components of either the FUSF and WUSF are indistinguishable from each other by invariantly defined properties on any infinite Cayley graph. Another crucial component of these proofs is the Mass-Transport Principle, which I will also give an introduction to.

Based in part on joint work with Asaf Nachmias.

### 11th April 2016 - Nic Freeman (University of Sheffield)

**Hybrid zones and mean curvature flow**

I will discuss the motion of hybrid zones, in the context of the Spatial Lambda-Fleming-Viot process. Hybrid zones are thin interfaces that form between populations which are genetically distinct, but still able to interbreed. The work relies on a new connection between branching Brownian motion, the Allen-Cahn equation and mean curvature flow.

### 18th April 2016 - Laurent Tournier (Paris 13)

**Random walks in Dirichlet environment**

We consider random walks in i.i.d. random environment on Zd. Among the distributions of the environment, the case of Dirichlet laws stands out due to its interpretation as a linearly reinforced walk and to several remarkable identities that lead to sharp results. During this walk, I will present the model and review the main tools and properties known so far, with a focus on aspects regarding asymptotic direction.

### 25th April 2016 - Hendrik Weber (University of Warwick)

**The dynamic Phi^4 model - Scaling limits and small noise behaviour**

In this talk I will discuss some recent progress on the dynamic \Phi^4 model, which is formally given by a non-linear stochastic PDE which is driven by space-time white noise. Due to the irregularity of the noise for spatial dimension d \geq 2 solutions are distribution valued and a renormalisation procedure has to be performed to interpret the non-linear term.

I will discuss how two situations in which such a renormalised solution appears naturally. First, I will discuss how it can be derived as a scaling limit of Ising-type models with a long range interaction. In this case the "infinite normalisation constant" has a natural interpretation as a shift of the inverse temperature. In the second part I will discuss the behaviour of solutions for small noise strength. I will argue that despite the infinite normalisation constant these renormalised PDE are the natural perturbation of the deterministic Allen-Cahn equation. I will illustrate this on the level of large deviations and then show that transitions between stable states of the deterministic dynamics are governed by an Eyring-Kramer type formula.

Based on joint work with N. Berglund (Orléans), G. Di Gesù (Paris), J.C. Mourrat (Lyon).

### 9th May 2016 - Hermann Thorisson (University of Iceland)

**Palm Theory and Shift-Coupling**

Palm versions w.r.t. stationary random measures are mass-stationary, that is, the origin is at a typical location in the mass of the random measure. For a simple example, consider the stationary Poisson process on the line conditioned on having a point at the origin. The origin is then at a typical point (at a typical location in the mass) because shifting the origin to the n:th point on the right (or on the left) does not alter the fact that the inter-point distances are i.i.d. exponential. Another (less obvious) example is the local time at zero of a two-sided standard Brownian motion.

In this talk we shall first consider mass-stationarity on the line and the shift-coupling problem of how to shift the origin from a typical location in the mass of one random measure to a typical location in the mass of another random measure. Applications include finding patterns like the Brownian bridge in the realisation of a Brownian motion, and also patterns in planar Brownian motion. If time allows we shall then extend the view beyond the line, moving through the Poisson process in the plane and d-dimensional space towards general random measures on groups.

### 16th May 2016 - Nadia Sidorova (UCL)

**Delocalising the parabolic Anderson model**

The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials (including Pareto potentials) it is localised at just one point. In the talk, we discuss a natural modification of the parabolic Anderson model on Z, where the one-point localisation breaks down for heavy-tailed Pareto potentials and remains unchanged for light-tailed Pareto potentials, exhibiting a phase transition at the Pareto parameter 2. This is a joint work with Stephen Muirhead and Richard Pymar.

### 23rd May 2016 - Lorenzo Taggi (TU Darmstadt)

**High temperature regime in spatial random permutations**

Spatial random permutations are implemented by probability measures on permutations of a set with spatial structure; we consider random permutations on a d-dimensional regular grid where points are either mapped to one of their nearest neighbors or to themselves. A parameter alpha encodes the tendency of points to be mapped to a neighbor; the larger is alpha, the stronger is the bias toward jumps. The model exhibits two different regimes that involve the length of cycles. As alpha is large, exponential decay of the cycle length is observed; as alpha is small, long cycles are observed and their length increases with the size of the grid. I will present recent progress made jointly with Volker Betz (TU Darmstadt) for the regime of large alpha and discuss conjectures suggested by numerical simulations.

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