UK Easter Probability Meeting 2018

* A workshop about: *

Random Dynamics and Other Recent Developments

*April 9 ^{th}-13^{th}, University Of Sheffield *

The invited talks will be in and around the minicourse areas:

Random Geometry, Stochastic Dynamical Systems, Random Reinforced Processes.

* Invited Speakers: *

**Márton Balázs** (University of Bristol)

** Jacobi triple product via the exclusion process**
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I will give a brief overview of very simple, hence maybe less investigated
structures in interacting particle systems: reversible product blocking
measures. These turn out to be more general than most people would think, in
particular asymmetric simple exclusion and nearest-neighbour asymmetric zero
range processes both enjoy them. But a careful look reveals that these two are
really the same process. Exploitation of this fact will give rise to the Jacobi
triple product formula - an identity previously known from number theory and
combinatorics. I will derive it from pure probability this time, and I hope to
surprise my audience as much as we got surprised when this identity first
popped up in our notebooks

**Elisabetta Candellero** (University of Warwick)

** Coexistence of competing first-passage percolation on hyperbolic graphs **
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We consider two first-passage percolation processes FPP_1 and FPP_{\lambda}, spreading with rates 1 and \lambda > 0 respectively, on a non-amenable hyperbolic graph G with bounded degree.
FPP_1 starts from a single source at the origin of G, while the initial con figuration of FPP_{\lambda} consists of countably many seeds distributed according to a product of iid Bernoulli random variables of parameter \mu > 0 on V (G)\{o}. Seeds start spreading FPP_{\lambda} after they are reached by either FPP_1 or FPP_{\lambda}. We show that for any such graph G, and any fixed value of \lambda > 0 there is a value
\mu_0 = \mu_0(G,\lambda ) > 0 such that for all 0 < \mu < \mu_0 the two processes coexist with positive probability. This shows a fundamental difference with the behavior of such processes on Z^d.
(Joint work with Alexandre Stauffer.)

**Sunil Chhita** (Durham University)

** A (2+1)-dimensional Anisotropic KPZ growth model with a rigid phase.**
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Stochastic growth processes in dimension (2+1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes known as the isotropic KPZ class and the anisotropic KPZ class (AKPZ). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class, fluctuations are logarithmic in time and space. These classes are determined by the sign of the determinant of the Hessian of the speed of growth.

It is natural to ask (a) if one can exhibit interesting growth models with "rigid" stationary states, i.e., with O(1) fluctuations (instead of logarithmically or power-like growing, as in Wolf's picture) and (b) what new phenomena arise when the speed of growth is not smooth, so that its Hessian is not defined. These two questions are actually related and in this talk, we provide an answer to both, in a specific framework. This is joint work with Fabio Toninelli (CNRS and Lyon 1).

**Matthias Hammer** (Technische Universität Berlin)

** Entrance laws for annihilating Brownian motions **
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Consider a system of particles moving independently on Brownian paths such that whenever two of them meet, the colliding pair annihilates instantly. The construction of such a system of annihilating Brownian motions (aBMs) is straightforward as long as we start with a finite number of particles, but is more involved for infinitely many particles. In particular, if we let the set of starting points become increasingly dense in the real line it is not obvious whether the resulting systems of aBMs converge and what the possible limit points (entrance laws) are.

In this talk, we will see that aBMs arise as the interface model of the continuous-space voter model. This link allows us to provide a full classification of entrance laws for aBMs. We also present some illustrating examples showing how different entrance laws can be obtained via finite approximations. Finally, we discuss a generalization to the infinite rate symbiotic branching model, of which the voter model is a special case.

Joint work with Marcel Ortgiese and Florian Völlering (University of Bath)

**Alexander E. Holroyd** (University of Washington)

** Polluted bootstrap percolation **
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Bootstrap percolation is a fundamental cellular automaton model for nucleation. Despite its simplicity, the model holds many surprises. I'll focus on how growth from sparse random seeds is affected by sparse random impurities in the medium. The answer will involve using recent oriented surface technology to construct a stegosaurus. Based on joint work with Janko Gravner and David Sivakoff.

**David Leslie** (University of Lancaster)

** Applied abstract stochastic approximation **
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I will introduce the main ideas involved in applying Benaim’s techniques to so-called “abstract stochastic approximation” in which the parameter of interest is an element of a function space. I will then demonstrate the method’s use to analyse learning in games where the action space is continuum. Under these scenarios the object of interest is a probability distribution over the continuum action space, and the noisy observations are samples from the evolving distribution. Joint work with Steven Perkins.

**Cécile Mailler** (University of Bath)

** Infinitely-many-colour urns by stochastic approximation.**
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Measure-valued Pólya processes (MVPPs) were introduced in 2017 as a generalisation of Pólya urns to infinitely-many colours. In this joint work with Denis Villemonais (Nancy), we exploit the link between Pólya urns and quasi-stationary distributions (already exhibited by Aldous, Flannery and Palacios in 1988) and use stochastic approximation techniques on a space of measures to prove almost sure convergence of a large class of MVPPs.

**Balázs Ráth** (Budapest University of Technology)

** The window process of slightly subcritical frozen percolation **
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The mean field frozen percolation process is a dynamic random graph
model which starts with the empty graph on N vertices, an edge
between a pair of vertices is added at rate 1/N and connected
components of size k are deleted at rate r * k, where r is a constant
that depends on N. This model is known to exhibit self-organized
criticality when 1 << N and 1/N << r << 1, see [Rath, Toth, 2009,
EJP], [Rath, 2009, JSP], i.e., the dynamics keep the graph in a state
which is essentially a near-critical critical Erdos-Renyi graph. One
defines the window process w(t) = A(t) * t / N, where A(t) is the
number of vertices alive at time t. We derive scaling limits for the
time evolution of w(t) when (1/N)^3 << r << 1, thus giving a detailed
picture of the mechanism that produces the self-organized criticality
of the model. Joint work with James Martin (Oxford) and Dominic Yeo
(Technion).

**Frank Redig** (Technische Universität Delft)

** Density fluctuations for the inclusion process in the condensation limit **
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We consider the symmetric inclusion process, which is a system of particles performing random walks on the lattice
Z^d and interacting by attracting each other. In the limit of low random walk jump rate, this process shows condensation phenomena, i.e.,
large piles of particles are created. We look at how condensates emerge from a homogeneous initial product measure (coarsening).
We study this via the density correlation function and show how it scales to a quantity related to the local time of sticky Brownian motion.
The starting point of this analysis is self-duality and an exact formula for the transition probabilities of two inclusion particles.
Joint work with Gioia Carinci (Delft), Cristian Giardina (Modena)

**Silke Rolles** (Technische Universität Munchen)

** Convergence of vertex-reinforced jump processes
to an extension of the supersymmetric hyperbolic nonlinear
sigma model **
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Sabot and Tarres discovered that the discrete time process
associated with the vertex-reinforced jump process is a
mixture of Markov chains. Surprisingly, the mixing measure
can be expressed in terms of a supersymmetric hyperbolic nonlinear
sigma model, which was introduced by Zirnbauer in a completely
different context.

In the talk, I will present an extension of Zirnbauer's model.
It is shown that it arises as a weak joint limit of a time-changed
version introduced by Sabot and Tarres of the vertex-reinforced
jump process. It describes the asymptotics of rescaled crossing
numbers, rescaled fluctuations of local times, asymptotic local
times on a logarithmic scale, endpoints of paths, and last exit trees.

This is joint work with Franz Merkl and Pierre Tarres

**Anja Sturm** (Universität Göttingen)

** On classifying genealogies for general (diploid) exchangeable population models **
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The genetic variation in a sample of individuals/genes depends on their relatedness
which is described by their genealogy. In this talk we consider classifying the genealogies
of exchangeable population models with fixed size N asymptotically as N tends to infinity.

In the first part of the talk we review classical results regarding the haploid Cannings model. Here, each individual
is represented by one of its genes and thus each offspring (gene) has a unique parent. In each generation,
the offspring vector to the N parents is exchangeable. With an appropriate rescaling the corresponding
coalescence processes describing the genealogy converge to a limit process. Möhle and Sagitov (2001) classified
all possible limit processes and showed that depending on the tail behavior of the offspring numbers the limit
process is Kingman's coalescent with coalescence of pairs or is be given by coalescents with (simultaneous)
multiple mergers in which (several) larger groups may find a common ancestor at the same time.

In the second part of the talk we extend this result to diploid bi-parental analogues of the Cannings model.
Here, the next generation is composed of offspring of parent pairs, which form an exchangeable (symmetric) array.
Also, every individual carries two gene copies, each of which is inherited from one of its parents. Our result
classifies the limiting coalescent processes describing the gene genealogies. Using this general result we determine
the limiting coalescent in a number of examples of which some have been studied previously (in special cases)
and some are new. We also point out connections to the theory of random graphs.

This talk is based on joint work with Matthias Birkner (Universität Mainz) and Huili Liu (Hebei Normal University).

**Amandine Véber** (École Polytechnique)

** The effects of a weak selection pressure in a spatially structured population **
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One of the motivations for the introduction of the Fisher-KPP equation was to model the wave of advance of a favourable (genetic) type in a population distributed over some continuous space. This model relies on the fact that reproductions occur very locally in space, so that if we assume that individuals can be of two types only, the drift term modelling the competition between the types is of the form sp_{t,x}(1-p_{t,x}). Here, s is the strength of the selection pressure and p_{t,x} is the frequency of the favoured type at location x and time
t. However, large-scale extinction-recolonisation events may happen at some nonnegligible frequency, potentially disturbing the wave of advance. In this talk, we shall address and compare the effect of weak selection in the presence or absence of occasional large-scale events, based on a model of evolution in a spatial continuum called the spatial
Lambda-Fleming-Viot process. This is a joint work with Alison Etheridge and Feng Yu.

**Stanislav Volkov** (Lund University)

** Border aggregation model **
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Consider a graph G with a subset of "sick" vertices called "the border". A particle released from the origin performs a random walk on G until it comes in the direct contact with a sick particle, at which point it becomes sick itself, and stops walking, thus increasing "the border" by one point. A new particle is then released from the origin, and the process repeats until the origin itself becomes a part of the border. We are interested in the total number xi of particles to be released by this final moment. Incidentally, this model can be viewed as a generalization of the OK Corral model.

We obtain distributions and bounds for xi when G is a star graph, a regular tree, and a d-dimensional lattice. (Based on the joint work with Debleena Thacker).

**Alex Watson** (University of Manchester)

** A probabilistic approach to spectral analysis of growth-fragmentation equations **
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The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this talk, I discuss a probabilistic approach to the study of this asymptotic behaviour. The method is based on the Feynman-Kac formula and a related martingale technique. This is joint work with Jean Bertoin.

* Contributed Talks: *

**George Andriopoulos** (University of Warwick)

** Convergence of blanket times for sequences of random walks on critical random graphs **
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Under the main assumption that sequences of graphs, associated measures, random walks, and local times converge in a suitable Gromov-Hausdorff sense, we prove distributional bounds for the blanket times of the random walks on the graphs in the sequence. We are able to establish convergence of the blanket times of the random walks on the largest component of the Erdős-Rényi random graph in the critical window, as well as convergence of the blanket times of the random walks on critical Galton-Watson trees. We demonstrate how our results can be used to establish convergence in a number of examples of critical random graphs such as the configuration model.

**Elia Bisi**(University of Warwick)

** Point-to-line last passage percolation via symplectic Schur functions **
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We study an exactly solvable model in the KPZ universality class, strongly connected to the totally asymmetric simple exclusion process (TASEP): last passage percolation with exponentially distributed waiting times, in the point-to-line path geometry. We show that its distribution can be expressed in terms of an integral of symplectic Schur functions. The rich algebraic structure behind this model permits deriving, in the scaling limit, Sasamoto’s Fredholm determinant formula for the GOE Tracy-Widom distribution.

Based on arXiv:1703.07337 and arXiv:1711.05120, joint work with Nikos Zygouras.

**Samuel Johnston ** (University College Dublin)

** The Coalescent Structure of Galton-Watson Trees**
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Take a continuous-time Galton-Watson tree. If the system survives until a large time T, then choose k particles uniformly from those alive. What does the ancestral tree drawn out by these k particles look like?

**Sandra Palau ** (University of Bath)

** Backbone and Spine decomposition for multi-type branching processes **
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We can construct different decomposition for multi-type continuous-state branching processes. In this talk we are going to focus in the backbone and in the spine decomposition. We will show the similarities and differences of them. Finally, we are going to provide some applications.

**Andreas Sojmark ** (University of Oxford)

** An SPDE model for Systemic Risk with Endogenous Contagion**
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We propose a dynamic model for the study of systemic risk in in large financial systems. The model is phrased as an interacting particle system with a common source of noise and a form of mean-reversion in the drift. Moreover, we introduce an endogenous contagion mechanism whereby the default of one institution can cause a drop in the financial health of the other institutions. In order to have a general model for systemic — or macroscopic — events, we show that the system converges to a unique mean field limit, characterized by a nonlinear SPDE on the half-line (with a Dirichlet boundary condition), which governs the conditional law of a ‘conditional McKean–Vlasov’ type diffusion. Depending on the realizations of the common noise and the strength of the mean reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, sparked by a devaluation of the common exposures, there are events of small probability that, through amplification by herd behaviour, can give rise to systemic default cascades. This is joint work with Ben Hambly (and closely related to joint work with Sean Ledger).

**Dominic Yeo ** (Israel Institute of Technology)

** Criticality in random transposition random walk**
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The random walk on the permutations of [N] generated by the transpositions was shown by Diaconis and Shahshahani to mix with sharp cutoff around N log N /2 steps. However, Schramm showed that the distribution of the sizes of the largest cycles concentrates (after rescaling) on the Poisson-Dirichlet distribution PD(0,1) considerably earlier, after (1+\epsilon)N/2 steps. We show that this behaviour in fact emerges precisely during the critical window of (1+\lambda N^{-1/3}) N/2 steps, as \lambda \rightarrow\infty. Our methods are rather different, and involve an analogy with the classical Erdos-Renyi random graph process, the metric scaling limits of a uniformly-chosen connected graph with a fixed finite number of surplus edges, and analysing the directed cycle structure of large 3-regular graphs. Joint work with Christina Goldschmidt.