Given a sequence of numbers p_n in [0,1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability p_n, n>1, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as n→∞? We show that a number of phase transitions take place as the turning gets slower (i.e. p_n is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is p_n=const/n. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws. The critical regime is particularly interesting: when the corresponding random walk is considered, an interesting process emerges as the scaling limit; also, a connection with Polya urns will be mentioned. This is joint work with S. Volkov (Lund) and Z. Wang (Boulder). See also the paper https://drive.google.com/file/d/0B4ZkCm_J6qB8NXNRcUFPM0hqRUU/view

The hard-core model has attracted attention for quite a long time; the first rigorous results about the phase transition on a lattice were obtained by Dobrushin in late 1960s. Since then, various aspects of the model gained importance in a number of applications. We propose a solution for the high-density hard-core model on a triangular lattice. The high-density phase diagram (i.e., the collection of pure phases) depends on arithmetic properties of the exclusion distance D; a convenient classification of possible cases can be given in terms of Eisenstein primes. For two classes of values of D the phase diagram is completely described: (I) when either D or D/√3 is a positive integer whose prime decomposition does not contain factors of the form 6k+1, (II) when D² is an integer whose prime decomposition contains (i) a single prime of the form 6k+1, and (ii) other primes, if any, in even powers, except for the prime 3. For the remaining values of D we offer some partial results. The main method of proof is the Pirogov-Sinai theory with an addition of Zahradnik's argument. The theory of dominant ground states is also extensively used, complemented by a computer-assisted argument.
This is a joint work with A. Mazel and Y. Suhov.

In this talk, we will establish a connection between random recursive tree, branching Markov chain and urn model. Exploring the connection further we will derive fairly general scaling limits for urn models with colors indexed by a Polish Space and show that several exiting results on classical/non-classical urn schemes can be easily derived out of such general asymptotic. We will further show that the connection can be used to derive exact asymptotic for the sizes of the connected components of a random recursive forest, obtained by removing the root of a random recursive tree.
This is a joint work with Debleena Thacker.

In recent years, the amount of available information has become so vast in certain fields of applications that it is infeasible or undesirable to carry out the computations on a single server. This has motivated the design and study of distributed statistical or learning methods. In distributed methods, the data is split amongst different administrative units and computations are carried out locally in parallel to each other. The outcome of the local computations are then aggregated into a final result on a central machine.
First, we will compare the theoretical properties of various (Bayesian) distributed methods proposed in the literature on the benchmark signal in Gaussian white noise model. Then we consider the limitations and guarantees of distributed methods in general under communication constraints on the same benchmark nonparametric model.
This is an ongoing joint work with Harry van Zanten.

The lace expansion is one of the primary tools for proving that probability models in high dimensions have mean field behaviour. I will explain the previous sentence by describing joint work in progress with David Brydges and Mark Holmes in which we develop a continuous time lace expansion. To motivate our methods I will introduce a class of n-component field theories that are generalizations of the Ising model of ferromagnetism. When n is zero, one, or two we are able to analyze these models. Our results for the case n=2 are new.

The uniform spanning forests (USFs) of infinite an infinite graph G are defined as infinite volume limits of uniform spanning trees on finite subgraphs of G. In this talk, I will describe how we use a new way of sampling the USF using the random interlacement process to compute various critical exponents governing the large-scale geometry of trees in the forest in a wide variety of “high-dimensional” graphs, including Z^d for d <= 5 and every bounded degree nonamenable graph. I will then sketch how this allows us to compute related exponents describing the geometry of avalanches in the Abelian sandpile model on the same class of graphs.

The vertex-reinforced jump process (VRJP) is a linearly reinforced random walk in continuous time. Reinforcing means that the VRJP prefers to jump to vertices it has previously visited; the strength of the reinforcement is a parameter of the model.
Sabot and Tarres have shown that the VRJP is related to a model known as the H22 supersymmetric hyperbolic spin model, which originated in the study of random band matrices. By making use of results for the H22 model they proved the VRJP is recurrent for sufficiently strong reinforcement. I will present a new and direct connection between the VRJP and hyperbolic spin models (both supersymmetric and classical), and show how this connection can be used to prove that the VRJP is recurrent in two dimensions for all reinforcement strengths.
This talk is about joint work with R. Bauerschmidt and A. Swann.

Let B denote the range of the Brownian motion in R^d. For a deterministic Borel measure ν we wish to find a random measure μ such that the support of μ is contained in B and the expectation of μ is ν. We discuss when exactly we can find such a μ.

Let B denote the range of the Brownian motion in R^d. For a deterministic Borel measure ν we wish to find a random measure μ such that the support of μ is contained in B and the expectation of μ is ν. We discuss when exactly we can find such a μ.

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+ε)N/2 steps. We show that this behaviour truly emerges precisely during the critical window of (1+λ N^-1/3) N/2 steps, as λ →∞. 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.

Let hard ball scatterers of radius r be placed in R^d, centred at the points of a Poisson point process of intensity ρ. The volume fraction r^dρ is assumed to be sufficiently low so that with positive probability the origin is not trapped in a finite domain fully surrounded by scatterers. The Lorentz process is the trajectory of a point-like particle starting from the origin with randomly oriented unit velocity subject to elastic collisions with the fixed (infinite mass) scatterers. The question of diffusive scaling limit of this process is one of the major open problems in classical statistical physics.
Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0, ρ→∞ so that r^d-1ρ→1 and the time scale is fixed, the Lorentz process (described informally above) converges to a Markovian random flight process, with independent exponentially distributed free flight times and Markovian scatterings. It is essentially straightforward to see that taking a second diffusive scaling limit (after the Gallavotti-Spohn limit) yields invariance principle.
I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3: Letting r→0, ρ→∞ so that r^d-1ρ→1 (as in B-G) and simultaneously rescaling time by T∼r^-2+ε we prove invariance principle (under diffusive scaling) for the Lorentz trajectory. (Note that the B-G limit and diffusive scaling are done simultaneously and not in sequel.) The proof is essentially based on control of the effect of recollisions by probabilistic coupling arguments. The main arguments are valid in d=3 but not in d=2.
Joint work (partly in progress) with Chris Lutsko (Bristol).

Let hard ball scatterers of radius r be placed in R^d, centred at the points of a Poisson point process of intensity ρ. The volume fraction r^dρ is assumed to be sufficiently low so that with positive probability the origin is not trapped in a finite domain fully surrounded by scatterers. The Lorentz process is the trajectory of a point-like particle starting from the origin with randomly oriented unit velocity subject to elastic collisions with the fixed (infinite mass) scatterers. The question of diffusive scaling limit of this process is one of the major open problems in classical statistical physics.
Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0, ρ→∞ so that r^d-1ρ→1 and the time scale is fixed, the Lorentz process (described informally above) converges to a Markovian random flight process, with independent exponentially distributed free flight times and Markovian scatterings. It is essentially straightforward to see that taking a second diffusive scaling limit (after the Gallavotti-Spohn limit) yields invariance principle.
I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3: Letting r→0, ρ→∞ so that r^d-1ρ→1 (as in B-G) and simultaneously rescaling time by T∼r^-2+ε we prove invariance principle (under diffusive scaling) for the Lorentz trajectory. (Note that the B-G limit and diffusive scaling are done simultaneously and not in sequel.) The proof is essentially based on control of the effect of recollisions by probabilistic coupling arguments. The main arguments are valid in d=3 but not in d=2.
Joint work (partly in progress) with Chris Lutsko (Bristol).