
Toronto Probability Seminar 200809
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:
probsematmathdottorontodotedu

2009

Speaker and Talk Title

Monday, April 6, 2009,
4:10 pm 
Emmanuel Schertzer
(Columbia)
The Voter Model and the Potts Model in One Dimension
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.

Monday, March 30, 2009,
4:10 pm 
Lionel Levine
(MIT)
Diamond Aggregation
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 oneparameter family
of walks of this type, which exhibit a phase transition in
the order of fluctuations.
Joint work with Wouter Kager.

Monday, March 23, 2009,
4:10 pm 
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
scaleinvariant 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 BaumslagSolitar groups $BS(1,m)$, and the affine
groups $\Z^d \rtimes GL(\Z,d)$ are all scaleinvariant.

Monday, March 16, 2009,
4:10 pm 
John Mayberry
(Cornell)
Evolution in Predator Prey Systems
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 BrunetDerrida type branchingselection particle
system.

Monday, March 9, 2009, 4:10 pm 
Ron Peled (NYU Courant)
Gravitational Allocation of Poisson Points

Monday, March 2, 2009,
4:10 pm 
Gidi Amir (University
of Toronto)
1Dimensional Long Range Diffusion Limited Aggregation(DLA)
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_{n1}, 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 firstorder 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.

Monday, February 9,
2009, 4:10 pm 
Grard Letac (Universit
Paul Sabatier, Toulouse)
The mean perimeter of some random plane convex sets generated
by a Brownian motion
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^{n1}
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.

Monday, February 2,
2009, 4:10 pm 
Tom Alberts, University of Toronto
Bridge Decomposition of Restriction Measures
In the early 60s Kesten showed that selfavoiding walk in
the upper half plane has a decomposition into an i.i.d. sequence
of "irreducible bridges". Loosely defined, a bridge
is a selfavoiding 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 2dimensional case, one can ask if the
(likely) scaling limit of selfavoiding 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.

Monday, January 26,
2009, 4:10 pm 
Gbor Pete (Toronto)
The scaling limits of dynamical and nearcritical percolation,
and the Minimal Spanning Tree
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 leftright 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 nearcritical 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.

Monday, January 19, 2009, 4:10 pm 
Manjunath Krishnapur
Limiting Spectral Distributions of NonHermitian Random Matrices 
Monday, January 12, 2009, 4:10 pm 
Senya Shlosman, (Lumini)
Phase transitions in systems with continuous symmetries 
2008

Speaker and Talk Title

Monday, November 29,
2008, 4:10 pm 
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.

Tuesday Oct. 14, 2008,
4:10 PM 
Gideon Amir (University
of Toronto)
The speed process of the Tottaly Assymetric Simple Exclusion
Process
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 multitype 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.

Tuesday Oct. 14, 2008,
1:00 PM
Room 230
*Note: Unusual Date and Time 
Jeff Steif, Chalmers
Institute of Technology, Sweden
Dynamical sensitivity of the infinite cluster in critical
percolation.
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 nth 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 nth 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.

Monday Sep 29, 2008,
4:30 PM
Stewart Library 
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 DoobMeyer decomposition theorem to construct a measurevalued
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 ``ddimensional''
transformation rule between domains. Finally it is shown that
mu, under some mild additional assumptions, is the unique
measurevalued function of SLE(kappa) curves that satisfies
a Domain Markov property arising from the transformation rule.

Monday Sep 15, 2008,
4:10 PM
Stewart Library 
Siva Athreya (Indian Statistical
Institute)
Survival of the contact process on the hierarchical group 
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