## abstracts for contributed talks

### Yoshihiro Abe, Kyoto University

Title: Effective resistances for supercritical percolation clusters in boxes

Abstract: Consider supercritical bond percolation on the lattice. In this talk, I will describe a result on effective resistances for the largest open cluster in a box. I will apply this to estimating the cover time and the maximum of the Gaussian free field on the largest cluster. These results imply that there are quantitative differences between the cluster and the lattice in three or higher dimensional cases.

### Erich Baur, ENS Lyon

Title: Exit laws of (an)isotropic random walks in random environment

Abstract: I will discuss exit laws from large balls of random walks in an i.i.d. random environment in Z^d, d>= 3, which is close to the fixed environment corresponding to simple random walk. Under a symmetry condition on the measure governing the environment, the exit laws approach those of a symmetric random walk, whose underlying nearest-neighbour transition kernel is a small perturbation of the simple random walk kernel charging each neighbour with mass 1/(2d)$. Joint work with Erwin Bolthausen.

### Yinshan Chang, MPI MIS in Leipzig

Title: Phase transition in loop percolation

Abstract: The talk is based on a recent joint work with Artem Sapozhnikov: http://arxiv.org/abs/1403.5687 We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied by Le Jan and Lemaire. It is a model with long range correlations with two parameters α and κ. The non-negative parameter α measures the amount of loops, and κ plays the role of killing on vertices penalizing (κ > 0) or favoring (κ < 0) appearance of large loops. In this talk, we will consider the integer cube lattice Zd for d ≥ 3 and κ = 0 for which there is a non-trivial phase transition. Interestingly, we observe a critical like behavior on the whole sub-critical domain of α, namely, for κ = 0 and any sub-critical value of α, the probability of one-arm event decays at most polynomially. We define a threhold corresponding to the finiteness of the expected cluster size which is non-trivial iff d ≥ 5. For d ≥ 5, below this threshold, we get, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For d = 3 or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, the one-arm exponent in dimension 3, if it exists, has to depend on the intensity α.

### Nicholas Cook, UCLA

Title: Random regular digraphs: Singularity and discrepancy

Abstract: We show that the adjacency matrix of a uniform random regular directed graph is invertible with high probability, assuming that the graph is dense. This is an important step toward proving this random matrix ensemble lies in the circular law universality class. The main challenge is to overcome the dependencies among the matrix entries. Our approach exploits local symmetries of the matrix distribution to ``inject" independent random variables into the problem. We also prove some discrepancy properties for the distribution of edges in the graph, which may be of independent interest.

### Owen Daniel, University of Warwick

Title: Loops, Soups, and Bosons

Abstract: In this talk we provide an introduction to Le Jan's model for Markovian loop soups, and their associated local field. In particular we describe Le Jan's isomorphism relating the local field of a symmetric loop soup to the Gaussian free field. In the second part of the talk we describe a particular class of non-symmetric loop soups, which we relate to the ideal Bose gas, and discuss how the isomorphism theorem can be used to elicit geometrical descriptions of classical critical phenomena. Joint work with Stefan Adams.

### Loic De Raphelis, Universite Paris 6 - LPMA

Title: Biased random walk on Galton Watson trees : coupled CLT of the height function and the track

Abstract: We consider a supercritical Galton-Watson tree conditioned to survive, and a nearest-neighbour random walk on this tree, with bias to the parent. In 2006, Y. Peres and O. Zeitouni proved a quenched central limit theorem on the height function of the random walk, that is the function giving the height of the walk in the tree with respect to time. The method introduced in the proof was later used by G. Faraud in 2008, and then A. Dembo and N. Sun in 2010 to extend this theorem to other objects. In this talk, we will introduce a new method which allowed us to prove these previous results an other way, with optimal conditions. This method will be based on the track of the walk, that is the set of vertices of the tree which have been visited by the walk. Moreover, we will give an explicit link between the track and the height function of the walk, stating a theorem on the convergence of their coupled law.

### Yuval Filmus, IAS

Title: Structure theorems for almost-linear functions on the symmetric group

Abstract: The classical Friedgut–Kalai–Naor theorem states that an almost linear function on the Boolean cube is close to a dictatorship. We discuss analogs of this theorem for functions in other domains: slices of the Boolean cube and the symmetric group. Joint work with David Ellis and Ehud Friedgut.

### Laura Florescu, NYU-Courant

Title: The range of a rotor walk and recurrence of directed lattices

Abstract: Rotor walk is a deterministic analogue of random walk introduced in 1996 in \cite{pddk} by Priezzhev, Dhar, Dhar and Krishnamurthy, and independently by Jim Propp \cite{propp}. In a \emph{rotor walk} on a graph, a particle exits a vertex in a predetermined cyclic fashion. We show a lower bound of $t^{d/d+1}$ on the number of sites visited by iid rotor walk in $t$ steps on any lattice in a $d$ dimensional space, thus proving the lower bound of $t^{2/3}$ conjectured for the square grid in \cite{pddk}. Additionally, we show that the range of rotor walk on the comb graph is also $t^{2/3}$, contrasting with that of random walk which is $\sqrt{t}\log{t}$. We also show recurrence of rotor walk on the Manhattan and F-lattices through connections to percolation arising from the 'stochastic pin ball' also known as the Lorentz wind-tree mirror model.

### Eric Foxall, UVic

Title: The SEIS model, or, the contact process with a latent stage

Abstract: The susceptible-exposed-infectious-susceptible (SEIS) model is well-known in mathematical epidemiology as a model of infection in which there is an average incubation time between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received little attention. Here we collect analytical results analogous to those obtained for the contact process, including bounds on critical values and a partial duality, and we identify the limiting behaviour in the limit of large incubation time. We also show convergence of critical values in the limit of small and large incubation time.

### Franck Gabriel, LPMA

Title: Yang-Mills measures.

Abstract: This talk will be a short overview of the resurgent theory of Yang-Mills measure : construction, characterization, convergence, and new paradigm for the theory of random matrices.

### Tyler Helmuth, UBC

Title: Loop Weighted Walk

Abstract: Loop weighted walk with weight $\lambda$ ($\lambda$-LWW) is a non-Markovian model of random walks that are discouraged (or encouraged, depending on $\lambda$) from completing loops: a walk receives a weight $\lambda^{n}$ if it contains $n$ loops. An important and challenging feature of this model is that it is not purely repulsive: the weight of the future of a walk may either increase or decrease if the past is forgotten. I will describe a representation of $\lambda$-LWW that enables a lace expansion analysis in high dimensions.

### Tobias Johnson, University of Southern California

Title: The frog model on trees

Abstract: The frog model is a self-interacting branching random walk defined like this: Put a sleeping frog at each vertex of a graph. At time 0, one of these frogs wakes up and begins a simple random walk in discrete time. Every frog it lands on wakes up and begins its own independent walk, also waking any sleepers it visits. The main result I'll present is a phase transition for transience and recurrence of the frog model on infinite $d$-ary trees. For $d=2$, the root is visited infinitely many times and all frogs wake up almost surely. For $d\geq 5$, the root is visited only finitely many times and some frogs never wake up. Joint work with Chris Hoffman and Matt Junge.

### Gursharn Kaur, Indian Statistical Institute

Title: Urn Schemes with Negative Reinforcements

Abstract: In this work we consider negatively reinforced Polya type urn processes with finitely many colors. We consider two types of such processes Linear and Inverse. We show that in case of only two colors both the processes are equivalent and the limiting configuration of the urn is uniform a.s., and for more than two color case the configuration of the urn after $n$ draws converge a.s. to a limiting distribution which is uniform if and only if uniform distribution is the unique stationary distribution of the underline Markov chain. "This is a joint work with my supervisor Antar Bandyopadhyay at Indian Statistical Institute,New Delhi".

### Alexander Kister, University of Warwick

Title: Pinning transition for fields with Laplacian interaction

Abstract: We study random fields given by an Hamiltonian favouring fields with zero curvature. Those models are used to model interfaces. We say that the interface is pinned if the interface gets an reward for being at zero. Depending on the strength of the reward and the boundary condition the interface either picks up the reward or not. We use large deviation techniques to study the phase transition. Joint work with Stefan Adams and Hendrik Weber.

### Brett Kolesnik, University of British Columbia

Title: Geodesic structure of the Brownian map

Abstract: The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed in the map, and with this, gain insight into its geodesic structure. Joint work with Omer Angel and Grégory Miermont.

### Piotr Nayar, University of Warsaw

Title: Kchinchine inequality for even moments with optimal constants

Abstract: We derive Khinchine type inequalities for even moments with optimal constants from the result of Walkup which states that the class of log-concave sequences is closed under the binomial convolution. This is a joint work with Krzysztof Oleszkiewicz.

### Kazuki Okamura, Graduate School of Mathematical Sciences, The University of Tokyo

Title: On the range of random walk on graphs satisfying a uniform condition

Abstract: We consider the range of random walks up to time n, R_n, on graphs satisfying a uniform condition. Not only all vertex transitive graphs but also many non-regular graphs satisfy the condition. We state certain weak laws of R_n from above and below. We also state that there is a graph such that it satisfies the condition and a sequence of the mean of R_n/n fluctuates.

### Moumanti Podder, Courant Institute of Mathematical Sciences, New York University

Title: Weighted Sums of Regularly Varying Random Variables with Dependent Weights

Abstract: \par Our work aims to study the tail behaviour of weighted sums of the form $$\sum_{t=1}^{\infty} X_t \prod_{j=1}^{t}Y_{j},$$ where each of the sequences $\{X_t\}$ and $\{Y_{t}\}$ is independent and identically distributed, with each $X_{t}$ having a regularly varying tail, and $(X_{t}, Y_{t})$ jointly following the bivariate Sarmanov distribution. With assumptions similar to those used by Denisov and Zwart (2007) imposed on these two sequences, and with certain suitably summable bounds similar to those proposed by Hazra and Maulik (2012), we try to explore the tail distribution of the random variables $\sup_{1 \leq k \leq n}\sum_{t=1}^{k} X_t \prod_{j=1}^{t}Y_{j}$ and $\sup_{n \geq 1}\sum_{t=1}^{n} X_t \prod_{j=1}^{t}Y_{j}$.\\ \par If time permits, I shall talk a little about how we can infer about the tail of a random variable $X$ when we are given that $XY$ has a regularly varying tail, where $Y$ has a sufficiently lighter tail compared to $X$ and $(X, Y)$ jointly follow bivariate Sarmanov distribution. %% References (remove if you do not use this) \renewcommand{\refname}{\normalsize{\textbf{References}}} {\footnotesize{\begin{thebibliography}{99} {\bibitem{bib1} D.~Denisov and B.~Zwart. \newblock On a theorem of Breiman and a class of random difference equations. \newblock \emph{J. Appl. Probab.}, 44\penalty0 (4):\penalty0 1031--1046, 2007. } {\bibitem{bib2} R.S.~Hazra and K.~Maulik. \newblock Tail behaviour of randomly weighted sums. \newblock \emph{Adv. Appl. Probab.}, 44\penalty0 (3):\penalty0 794--814, 2012. } \end{thebibliography}}}

### Miklos Racz, UC Berkeley

Title: Coexistence in preferential attachment networks

Abstract: We introduce a new model of product adoption that focuses on word-of-mouth recommendations. Specifically, when a new node joins the network, it chooses neighbors according to preferential attachment, and then chooses its type based on the number of initial neighbors of each type. This can model, e.g., a new cell-phone user choosing a cell-phone provider. The main qualitative feature of our model is that often several competitors will coexist, which matches empirical observations in many current markets. This is joint work with Tonci Antunovic and Elchanan Mossel.

### Mustazee Rahman, University of Toronto

Title: Local algorithms for independent sets in random graphs

Abstract: An independent set in a graph is a set of vertices such that there are no edges between them. How large can an independent set be in a random d-regular graph? How large can it be if we are to construct it using a (possibly randomized) algorithm that is local in nature? We will discuss a recently introduced notion of local algorithms for combinatorial optimization problems on large, random d-regular graphs. We will then explain why, for asymptotically large d, local algorithms can only produce independent sets of size at most half of the largest ones. The factor of 1/2 turns out to be optimal. Joint work with Balint Virag.

### Michele Salvi, TU Munich

Title: Homogenization in the Random Conductance Model: A CLT for the Effective Conductance

Abstract: In nature, most materials have a rather complicated microscopic structure. Nevertheless, the global properties of the material (such as heat or electric conduction) can be described by differential equations with smooth coefficients. An explaination to this behavior is offered by Homogenization Theory: The rapid oscillations of local coefficients caused by impurities average out, or homogenize, at a macroscopic scale. The Effective Conductance of random electrical networks, i.e. the minimum of the Dirichlet Energy, is a prominent example of a homogenizing quantity. When scaled by the volume of the domain, the Effective Conductance was already known in the 80s to have an almost sure constant limit, but the order of fluctuations has remained an open problem for almost thirty years. In our work, we were able to derive a (non-degenerate) Central Limit Theorem for this quantity, albeit under rather restrictive hypothesis (i.i.d. conductances on the network with small ellipticity contrast, square domains and linear boundary conditions). The proof is based on the corrector method and the Martingale Central Limit Theorem, while a key integrability condition is furnished by the Meyers estimate. (joint work with Marek Biskup and Tilman Wolff)

### Marta Strzelecka, University of Warsaw

Title: Sharp ratio inequalities for martingales

Abstract: We will show the sharp ratio inequality $$ E \left(\frac{|M_t|} {s+\langle M \rangle _t } \right)^p \leqslant E\frac{C(p,s,D)}{(s+\langle M\rangle _t)^{p/2} }+D$$ for any continuous martingale $( M_t ) _{t\geqslant 0}$ (here $s>0$, $D\in \mathbb{R}$ and $p\geqslant 1$ are fixed). This is an analogue of an inequality proven in [1] for the case $p=q$, when the estimate from [1] does not hold. Our proof will be based on optimal stopping methods. We will find a special function, which is a solution to the heat equation in a class of functions satisfying some homogeneity property. In the case $p=1$ we will also see how the inequality works for sums of conditionally symmetric random variables. References [1] M. E. Caballero B. Fernandez and D. Nualart, Estimation of Densities and Applications, J. Theoret. Probab. 11 (1998), 831–851.

### Tomasz Tkocz, University of Warwick

Title: On some strange operators acting on L_1

Abstract: We shall discuss existence of bounded, linear operators acting on L_1[0,1] which are locally but not globally invertible (i.e. they are isomorphisms when restricted to the set of functions with small support, and yet they have infinite dimensional kernel). Joint work with Ball, Johnson, Nasseri and Schechtman.

### Alex Tomberg, UBC

Title: Logarithmic corrections to scaling for the 4 dimensional weakly self-avoiding walk: watermelon networks

Abstract: We calculate the logarithmic correction to the decay of the critical two-point function for networks of p mutually-avoiding weakly self-avoiding walks joining two distant points on the 4-dimensional integer lattice. While similar results have been obtained previously for dimensions d > 4 by lace expansion, our proof is based on a rigorous renormalisation group analysis of a representation of the self-avoiding walk as a supersymmetric field theory. The talk is based on joint and ongoing work with Roland Bauerschmidt and Gordon Slade.

### Florian Völlering, University of Göttingen

Title: Talagrand's inequality for Interacting Particle Systems

Abstract: Talagrand's inequality for independent Bernoulli random variables is extended to many interacting particle systems (IPS). The main assumption is that the IPS satisfies a log-Sobolev inequality. Additionally we also look at a common application, the relation between the probability of increasing events and the influences on that event by changing a single spin.

### Tobias Wassmer, University of Vienna

Title: Randomly trapped random walks on Zd

Abstract: We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on Zd, d\geq 2. Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has a slowly varying return probability, we show that the scaling limit of its clock process is either deterministic linearly growing or a stable subordinator. In the case when the discrete skeleton is a simple random walk on Zd, this implies that the scaling limit of the randomly trapped random walk is either Brownian motion or the Fractional Kinetics process, as conjectured in [BCCR13].