


Probability
Org: Martin Barlow and Edwin Pekins (UBC) [PDF]
 JUAN ALVAREZ, University of Saskatchewan, 142 McLean Hall, 106 Wiggins
Road, Saskatoon, SK S7N 5E6
Morita approximations to random copolymer localization
[PDF] 
We look at Morita approximations to analyze Dyck path models of random
copolymer localization at an interface between two immiscible liquids.
A polymer can be either delocalized, with zero density of monomers at
the interface, or localized, with a nonzero density of monomers at
the interface.
The phase diagram for the first order Morita approximation, which
constrains the number of one type of comonomers, consists of four
distinct regions: two delocalized regions, a localized region, and a
region of coexistence of the two delocalized regions.
It is known that the boundaries of the delocalized regions do not
change as the order of the Morita approximation is increased. It is
also known that the region of coexistence of the two delocalized
regions is not present in the quenched model, it is a localized region
instead. We have some numerical evidence that points toward the
existence of the region of coexistence of the two delocalized regions
for Morita approximations of up to eighth order. We also have some
arguments that point toward localization in the region of coexistence
of the two delocalized regions, for Morita approximations of twelfth
order and above.
We get tighter upper bounds on the average constrained free energy as
the order of the Morita approximation increases.
 OMER ANGEL, UBC
1 Dimensional DLA
[PDF] 
Diffusion limited aggregation (DLA) in 2 or more dimensions is an
infamously difficult model for the growth of a random fractal. DLA
was introduced in 1981 and attracted massive attention (63,400 Google
results for "diffusion limited aggregation"). Kesten's 1987 upper
bound 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 in complex ways. In particular there are three
phase transitions in the behaviour when the step distribution has
finite expectation, finite variance, and finite third moment.
Joint work with Gidi Amir, Itai Benjamini and Gadi Kozma.
 BALINT VIRAG, University of Toronto
Noise limits for complex eigenvalues
[PDF] 
Consider a Coulomb gas in the plane: n points distributed with
density proportional to product x_{i}x_{j}^{2} with respect to some
underlying measure m.
When m is Gaussian, this process coincides with the eigenvalues of
matrix filled with i.i.d. complex normal entries. Other choices of
m yield the zeros of i.i.d. Gaussian power series, the projection
of the spherical gas, or the eigenvalues of a random unitary matrix.
When the points get denser, these processes converge to noises that
are different from the usual white one.
This is joint work with Brian Rider.
 DAVID BRYDGES, University of British Columbia
Joint density for the local times of continuoustime random
walks
[PDF] 
We investigate the local times of continuoustime random walks on
arbitrary discrete state spaces. For fixed finite range of the random
walk, we derive an explicit formula for the joint density of all local
times on the range, at any fixed time. We apply this formula in the
following directions:
(1) we prove the analog of the wellknown RayKnight
description of Brownian local times for continuoustime simple
random walk on Z,
(2) we derive large deviation estimates for exponential
integrals of the normalized local times beyond the exponential
scale.
This work was done in collaboration with Remco van der Hofstad and
Wolfgang Koenig.
 JEANFRANCOIS DELMAS, ENPC, 68 av. Blaise Pascal, 77455 Champs sur Marne,
France
Fragmentation associated to continuous random trees
[PDF] 
This is a joint work with R. Abraham. There is a natural way to
consider continuous limit of GaltonWatson trees, using Lévy
process with no negative jumps (see the work of T. Duquesne, J.F. Le
Gall and Y. Le Jan). Each node of the continuous random tree has a
mass corresponding to the "number" of its children. We cut each
node at independent exponential random time with parameter equal to
the mass of the node. This procedure gives a collection of smaller
and smaller subtrees. Considering the evolution of the sequence of
the sizes of the subtrees as time goes on, we get a fragmentation
process. In the stable case, this corresponds to the selfsimilar
fragmentation described by G. Miermont. In the general case, we get a
non selfsimilar fragmentation process. We compute for the general
fragmentation process a family of dislocation measures as well as the
law of the size of a tagged fragment.
 ANTAL JARAI, Carleton University, Ottawa, ON
A selforganized critical forest fire model
[PDF] 
We study a simple forest fire model introduced by physicists Drossel
and Schwabl. In a certain limiting case, the model is expected to
behave similarly to critical statistical physics systems. Extensive
simulation results are available in the literature, but hardly
anything seems to be known rigorously, especially in dimensions two
and higher. We analyze the onedimensional model, which already shows
quite nontrivial behaviour. We are able to give bounds on the
cluster size distribution that suggest that this distribution shows
interesting behaviour on two separate scales. For "small" clusters
we expect a power law, and for "large" clusters another nontrivial
distribution.
Joint work with Jacob van den Berg.
 RICHARD KENYON, UBC, Vancouver, BC
Groves, Pfaffians, and crossing probabilities
[PDF] 
A planar essential spanning forest is a spanning forest of a planar
graph in which every component touches the outer boundary. The
components of such a spanning forest define a partition of the
boundary vertices. Given a random planar essential spanning forest of
a finite planar graph with n boundary vertices, we show how to
compute the probability of any particular partition as a rational
function of the pairwise resistances.
Groves, defined by Carroll and Speyer, are certain kinds of essential
spanning forests on planar graphs which arose in their study of the
"cube recurrence relation". We show that a certain class of
topologies of essential spanning forests, including groves, can be
computed using Pfaffians. The proof is a generalization of the
KarlinMcGregorLindstromGesselViennot method.
This is joint work with David Wilson.
 MIKE KOZDRON, University of Regina
Looperased random walk and Fomin's identity
[PDF] 
In this talk, I will outline the recent proof of a 2001 conjecture of
S. Fomin concerning a relationship between crossing probabilities of
twodimensional looperased random walk and planar Brownian motion.
This conjecture was proved by establishing estimates of random walk
exit probabilities in simply connected domains. These estimates can
also be used to give a quick derivation of a crossing exponent for
looperased random walk first proved by R. Kenyon in 2000.
This talk is based on joint work with Greg Lawler.
 VLADA LIMIC, University of British Columbia, Vancouver, BC V6T 1Z2
Recent progress on the attracting edge problem
[PDF] 
Reinforcement is observed frequently in nature and society, where
beneficial interactions tend to be repeated. An edge reinforced random
walker on a graph remembers the number of times each edge was
traversed in the past, and decides to make the next random step with
probabilities favouring places visited before. Using martingale
techniques and comparison with the generalized Urn scheme, it can be
shown that the edge reinforced random walker on a graph of bounded
degree, with the reinforcement weight function W(k) = k^{r},
r > 1, traverses a random attracting edge at all large
times, with probability 1. A remarkably short argument of Sellke
(1994) shows that attracting edge exists if and only if
whenever the underlying graph has no odd cycle. The conjecture that the
above summability condition implies existence of attracting edge when
the underlying graph is a triangle is still open. Progress has
been made recently towards better understanding of attracting edge
property for a large class of weights W satisfying the above
condition.
 JEREMY QUASTEL, University of Toronto
Travelling fronts in the stochastically perturbed KPP
equation
[PDF] 
We study the asymptotic speed of travelling fronts in a
reactiondiffusion equation perturbed by a FisherWright white
noise. We prove part of a conjecture of Brunet and Derrida that such
perturbations have an unusually strong effect on the front speed.
This is joint work with Carl Mueller (Rochester) and Leonid Mytnik
(Technion).
 BRUCE REED, McGill University, Montreal
On the diameter of a random Minimum Weight Spanning Tree
[PDF] 
In first passage percolation, given random weights on the edges of a
connected graph G, we are interested in the structure of the graph
formed by the union of the lightest paths from some source s to
every other vertex. A natural variant would be to consider the
lightest graph containing a path between every pair of vertices (as we
will see this makes sense even in the infinite setting). These
objects arise naturally in computer science and network design and are
called Minimum Weight Spanning Trees.
In such a tree, there is a unique path between every pair of vertices.
We are interested in the maximum number of edges in such a path, which
is known as the diameter of the tree. We consider weighting
the edges of a complete graph (the mean field model) with i.i.d. weights, conditioning on the weights being distinct (a probability 1
event if our distribution has no point mass).
Amongst other results on its distribution, we show that for any
weighting, the expected value of the diameter for the MWST on a
complete graph with n vertices is between c_{1} n^{[ 1/3]} and c_{2}n^{[ 1/3]} for two constants c_{1} and c_{2}.
This is joint work with Louigi AddarioBerry at end.
 TOM SALISBURY, York University and the Fields Institute
Singular stochastic integral equations
[PDF] 
I will describe work in progress concerning YamadaWatanabe type
singular solutions to stochastic integral equations. Complete results
in this direction would establish pathwise uniqueness for SPDE models
analogous to catalytic super Brownian motion. Unfortunately we don't
have complete results, but I will try to describe what progress has
been made to date.
This is joint work with Leonid Mytnik (Technion).
 CHRIS SOTEROS, University of Saskatchewan, 106 Wiggins Rd., Saskatoon, SK
Selfavoiding walk models of random copolymers
[PDF] 
Selfavoiding walk models have been used to study linear polymers
(long chain molecules) in dilute solution for about 50 years. For
such models, the vertices of the walk represent the monomer units
which compose the polymer and an edge of the walk joins two monomer
units which are chemically bonded together in the polymer chain.
Distinct selfavoiding walks on a lattice, such as the square or
simple cubic lattice, represent distinct conformations of the polymer
chain. Recently there has been much interest in extending the
standard selfavoiding walk models of homopolymers (all monomer units
considered identical) to the study of random copolymers. A random
copolymer is a polymer composed of k types of comonomers,
A_{1},...,A_{k}, where the specific sequence of these comonomers that
appear along the polymer chain is determined by a random process. The
monomer sequence can be thought of as being determined in or by the
polymerization process (assumed to involve a random process) but then
once determined the sequence of monomers is fixed; this is an example
of what is known as quenched randomness. In the simplest
selfavoiding walk model of a random copolymer, one assumes that k=2
and that the monomer sequence c_{1},c_{2},...,c_{n} (for a walk
of length n) is such that the c_{i}'s are independent and
identically distributed with P(c_{i}=A_{1})=p. In this talk, I will
review the progress that has been made using selfavoiding walk models
to study phase transitions, such as the adsorption phase transition
and the localization phase transition, in random copolymer systems.

