SCIENTIFIC PROGRAMS AND ACTIVITIES

Toronto Probability Seminar 2008-09
held at the Fields Institute

Organizers
Bálint Virág , Benedek Valkó
University of Toronto, Mathematics and Statistics

For questions, scheduling, or to be added to the mailing list, contact the organizers at:
probsem-at-math-dot-toronto-dot-edu

Speaker and Talk Title

The voter model can be seen as a simple model for describing the propagation of opinions in a population where neighbors influence each other. More precisely, every integer is assigned with an original opinion at time t=0 and then updates its opinion by taking on the opinion of one of its neighbors chosen uniformly at random with rate 1. In the first part of the talk, I will show that such a model can easily be described in terms of a system of coalescing random walks. In the second part of the talk, I will introduce a variation of the preceding model where the voters do not only change their mind under the influence of their environment, but where they are also able to come up with an opinion differing from their neighbors. This model is closely related to a classical model in statistical physics called the one dimensional stochastic Potts model. I will show that under the appropriate scaling, this model converges to a continuum object which can be constructed by a marking procedure of a family of coalescing Brownian motions.

Joint work with C. Newman and K. Ravishankar.

Start with n particles at the origin in Z^2, and let each perform a simple random walk until it reaches an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting set of n occupied sites is close to a disk. The order of fluctuations from circularity remains an open problem. I'll describe a way of modifying slightly the law of the walk so that the limiting shape becomes a diamond instead of a disk. There is a natural one-parameter family of walks of this type, which exhibit a phase transition in the order of fluctuations.

Joint work with Wouter Kager.

Gbor Pete (Toronto)
Random walks on percolation clusters and percolation renormalization on groups.

We show that for all $p > p_c(\Z^d)$ percolation parameters, the probability that the cluster of the origin is finite but is adjacent to the infinite cluster with at least $t$ edges is exponentially small in $t$. This result yields a simple proof that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice, which implies that simple random walk on the cluster behaves the same way. The same results hold for all finitely presented groups if $p$ is close enough to 1, but renormalization can be used on $\Z^d$ to get the full result.

We also examine the possibility of renormalization on other groups. Itai Benjamini conjectured that if a group $G$ is scale-invariant in the sense that has a finite index subgroup chain $G = G_0 > G_1 > G_2 > \dots$ with $G_i\simeq G$ and $\bigcap_i G_i=\{1\}$, then it has to be of polynomial growth. In joint work with V. Nekrashevych, we have given several
counterexamples: the lamplighter group $\Z_2 \wr \Z$, the solvable Baumslag-Solitar groups $BS(1,m)$, and the affine groups $\Z^d \rtimes GL(\Z,d)$ are all scale-invariant.

We shall discuss the adaptive dynamics of predator prey systems modeled by a dynamical system in which the characteristics are allowed to evolve by small random mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit prey coexistence and the parameters of the resident prey type converge to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. Depending on the parameters being varied we see (i) the number of coexisting predators remains tight and the differences of the parameters from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity and we can study the evolving process of coexisting predator characteristics via connections with killed branching random walks and a Brunet-Derrida type branching-selection particle system.

Diffusion limited aggregation (DLA) in 2 or more dimensions is an infamously difficult model for the growth of a random fractal. In the model, a sequence of aggregates A_n is built on the square lattice, by starting with a single point A_0={0}, and adding one particle at each step.The position of the particle added at step n is chosen by starting a simple random walk from "infinity" (far away) and letting the walk wander until it becomes a neighbour of the current aggregate A_{n-1}, at which time it is stopped and added to the aggregate to form A_n.

DLA was introduced in 1981 and attracted massive attention. (184,000 google hits). Even so, Kesten's 1987 upper bound on the diameter growth rate is almost the only proven result on it.

We define a variation of DLA in one dimension. This becomes interesting when the random walk generating the DLA has arbitrary long jumps. It turns out that the growth rate of the aggregate depends on the step distribution and more specifically on the decay of the tail opf the undrlying random walk. In particular we show that there are at least three phase transitions in the behaviour when the step distribution has finite 1/2 moment, finite variance, and finite third moment. And more suprisingly that there seems to be no first-order phase transition when the walk goes from the transient to the recurrent regimn (finite expectation).

If time permits, we will also discuss some results on the limit aggregate A_infinity, and show a transient random walk for which the aggregate eventually spans all points in Z.

Joint work with Omer Angel, Itai Benjamini and Gadi Kozma.

If C_1 is the convex hull of the curve of the standard Brownian motion in the complex plane watched from time 0 to 1, and if w is an nth root of unity, we consider the convex hull C_n of C_1 \cup w C_1 \cup w^2 C_1 \cup \ldots \cup w^{n-1} C_1.

For instance C_2 is the symmetrized convex hull of the Brownian curve. We compute the means of the perimeters of C_1, C_2, C_4 by elementary calculations as well as some other simple convex hulls. The computation of the means of the perimeter of C_3 and C_6 is more involved and is done by the computation of the distribution of the exit time by the standard Brownian motion of the fundamental domain for symmetry groups in Euclidean spaces.

Joint work with Philippe Biane.

Tom Alberts, University of Toronto
Bridge Decomposition of Restriction Measures

In the early 60s Kesten showed that self-avoiding walk in the upper half plane has a decomposition into an i.i.d. sequence of "irreducible bridges". Loosely defined, a bridge is a self-avoiding path that achieves its minimum and maximum heights at the start and end of the path (respectively), and it is irreducible if it contains no smaller bridges.
Considering only the 2-dimensional case, one can ask if the (likely) scaling limit of self-avoiding walk, the SLE(8/3) process, also has such a decomposition. I will talk about recent work with Hugo Duminil from Ecole Normale Superieure that provides a positive answer, using only the restriction property of SLE(8/3). In the end we are able to decompose the SLE(8/3) path as a Poisson Point Process on the space of irreducible bridges, in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros. Our decomposition can actually be generalized beyond SLE(8/3) and applied to an entire family of "restriction measures", hence the title of the talk. If time permits I will also talk about the natural time parameterization for SLE(8/3), which has immediate applications towards the bridge decomposition.

Let each site of the triangular lattice, with small mesh Q$\eta$, have an independent Poisson clock with a certain rate $r(Q\eta) = \eta^{3/4+o(1)}$ switching between open and closed. Then, at any given moment, the configuration
is just critical percolation; in particular, the probability of a left-right open crossing in the unit square is close to 1/2. Furthermore, because of the scaling, the expected number of switches in unit time between having a crossing or not is of unit order.

We prove that the limit (as $\eta \to 0$) of the above process exists as a Markov process, and it is conformally covariant: if we change the domain with a conformal map $\phi(z)$, then time has to be scaled locally by $|\phi'(z)|^{3/4}$. The same proof yields a similar result for near-critical percolation, and it also shows that the scaling limit of (a version of) the Minimal Spanning Tree exists, it is invariant under translations, rotations and scaling, but *probably* not under general conformal maps.

Joint work with Christophe Garban and Oded Schramm.

Speaker and Talk Title

Pierre Nolin (Courant institute)
Universality of some random shapes: inhomogeneity and SLE(6)

The physicists Gouyet, Rosso and Sapoval introduced in 1985 a model of inhomogeneous medium, known as "Gradient Percolation", to show numerical evidence that diffusion fronts are fractal. They measured the dimension
7/4, which can be observed in many other situations. We will discuss how one can prove mathematically the appearance of "universal" random shapes related to SLE(6) when some inhomogeneity - a density gradient - is
present. In particular we will show that fractal interfaces of dimension 7/4 spontaneously arise.

We study the exclusion process on Z where each particle is assigned a class (number in Z) and each particle tries to swap places with its right neighbour with rate 1 if that neighbor has a higher class number. (Alternatively each edge of Z is "sorted" with rate 1). With the right starting conditions, the position of each particle(Normalized by the time) converges to a constant speed. The speed of each particle is uniform in [-1,1], but there are strong dependencies between the behaviour of different particles. We study this exclusion process and the distribution of its related speed process. In particular we show the exsistence of infinite "convoys" - particles (with different classes) all converging to the same speed. We also give some new symmetries for the multi-type TASEP. Some of our results apply to the partially asymmetric case as well.

This is joint work with Omer Angel and Benedek Valko (until recently from Uof
T, now at UBC and university of Wisconsin)

All definitions will be given in the lecture. No prior knowledge of exclusion
processes is assumed.

We look at dynamical percolation in the case where percolation occurs at criticality. For spherically symmetric trees, if the expected number of vertices at the n-th level connecting to the root is of the order n(log n)^\alpha, then if \alpha > 2, there are no exceptional times of nonpercolation while if is in (1,2), there are such exceptional times. (An older result of R. Lyons tells us that percolation occurs at a fixed time if and only if \alpha >1.) It turns out that within the regime where there are no exceptional times, there is another type of ``phase transition'' in the behavior of the process. If the expected number of vertices at the n-th level connecting to the root is of the form n^\alpha, then if \alpha > 2,the number of connected components of the set of times in [0,1] at which the root is not percolating is finite a.s. while if \alpha is in (1,2), then the number of such components is infinite with positive probability. This is joint work with Yuval Peres and Oded Schramm.

Tom Alberts, University of Toronto
Dimension and Measure of SLE on the Boundary

In the range 4 < kappa < 8, it is well known that the intersection of a chordal SLE(kappa) curve with the real line is a highly irregular fractal set with Hausdorff dimension between zero and one. In this talk I describe the dimension and measure of this set. There are two main parts. In the first part the Hausdorff dimension is proven to be almost surely d := 2 - 8/kappa. This is done by using various tools from the theory of conformal mappings to derive an asymptotic upper bound on the probability that two disjoint intervals on the real line are hit by the curve, as the interval widths go to zero. In the second part an abstract appeal is made to the Doob-Meyer decomposition theorem to construct a measure-valued function mu of the curve that is almost surely supported on the intersection of the curve with the line. The measure gives a local description of the structure of the set that provides much finer information than just the Hausdorff dimension. Properties of the measure are then derived, along with a ``d-dimensional'' transformation rule between domains. Finally it is shown that mu, under some mild additional assumptions, is the unique measure-valued function of SLE(kappa) curves that satisfies a Domain Markov property arising from the transformation rule.