## Pacific Northwest Probability Seminar

- Start Date: 10/19/2013
- End Date: 10/20/2013

University of Washington

The Scientific Committee for the NW Probability Seminar 2011 consists of Omer Angel (U British Columbia), Chris Burdzy (U Washington), Zhenqing Chen (U Washington), Yevgeniy Kovchegov (Oregon State U), David Levin (U Oregon) and Eyal Lubetzky (Microsoft), Yuval Peres (Microsoft) and David Wilson (Microsoft).

**Day 1 (University of Washington) **

On day 1, the talks will take place in Savery Hall 260. See
the **map**
the location of Savery Hall and Padelford Hall (the Department of Mathematics
is in the Padelford Hall). More **campus maps** are available
at the UW Web site.

Parking on UW campus is free on Saturdays only after noon.
See **parking
information**.

**Schedule **

- 10:30
**Coffee**Savery 260 - 11:00
- 11:50
**Zhenqing Chen**, University of Washington

**Quenched Invariance Principles for Random Walks in Random
Media with Boundary **

In this talk, I will present recent progress on the study of quenched
invariance principles for random walks in random environments with a boundary.
In particular, we prove that the random walk on a supercritical percolation
cluster or amongst random conductances bounded uniformly from below in a
half-space, quarter-space, etc., converges when rescaled diffusively to a
reflecting Brownian motion, which had been one of the important open problems
in this area. Our approach uses a Dirichlet form extension theorem and
two-sided heat kernel estimates. Based on a recent joint work with David
Croydon and Takashi Kumagai.

- 12:00
- 2:00
**Lunch**, catered, HUB 250 (Husky Union Building), including remarks on Bob Blumenthal's contributions to probability by Ron Getoor. - 2:00 -
2:50
**Patrick Fitzsimmons**, University of California, San Diego

**"Birnbaum Lecture": The Stationary Process
Associated with an Excessive Measure of a Markov Process; Applications Old and
New **

I will discuss the stationary process with random times of birth and death that
can be associated with a given strong Markov process and one of its excessive
measures. (In the literature this process is commonly referred to as a
``Kuznetsov process''; the notion is related the ``quasi-process'' of G.A. Hunt
and M. Weil.) I will then examine a number of examples illustrating the
usefulness of this process, both as a conceptual device and as a technical
tool. These examples will involve applications to Skorokhod stopping, excursion
theory, time reversal, capacities, as well as a very recent proof of Kac's
scattering length formula.

- 3:00 -
3:50
**David Koslicki**, Oregon State University

**Random substitutions, Martin boundaries, and molecular
evolution **

In this talk, I will define the concept of a "substitution Markov
chain" (SMC) which can be conceptually viewed as a kind of random
substitution process. A number of properties will be shown, including the
computation of the Martin boundary in a few special cases. As an application,
it will also be shown that a certain class of SMC's can be used to accurately
model molecular evolution.

- 4:00 -
4:30
**Coffee break**Savery 260 - 4:30 -
5:20
**Balázs Ráth**, University of British Columbia

**Critical mean field frozen percolation and the
multiplicative coalescent with linear deletion **

The mean field frozen percolation process is a modification of the dynamical
Erdos-Renyi random graph process in which connected components are deleted
(frozen) with a rate linearly proportional to their size. It is known that the
model exhibits self-organized criticality for a wide range of choices of the
freezing rate. The aim of the present talk is to describe the results of our
upcoming joint paper with James Martin (Oxford) in which we show that a careful
choice of the freezing rate forces the graph to stay in the critical window
permanently. In particular, we find that the scaling limit of the evolution of
big component sizes is a variant of Aldous' multiplicative coalescent process.

- 6:30
No-host
**dinner**.- Restaurant:
**Salmon House**, 401 NE Northlake Way, Seattle, WA 98105 **Menu****Driving directions**from Padelford parking lot- We
will
**go Dutch**, that is, every person will pay for what he/she orders; the entree prices vary significantly.

- Restaurant:

**Day 2 (Microsoft Research) **

On day 2, the talks will take place in Building 99 at Microsoft. Parking at Microsoft is free.

Directions: From the north: Take I-5 south, then I-405
south, then WA-520 east.

From the south: Take I-5 north, then I-405 north, then WA-520 east.

From Seattle: Take WA-520 east.

By airplane: Fly to Seattle's airport, take I-405 north, then WA-520 east.

From WA-520 east, take the 148th Ave NE North exit (this is
the second 148th Ave NE exit). Turn right (north) onto 148th Ave NE, proceed a
few blocks, and turn right onto NE 36th St. Building 99 will be on the left.
The address is 14820 NE 36th St, Redmond, WA 98052-5319. **Click
here for a map**.

**Schedule **

- 9:30
**Coffee** - 10:00
- 11:00
**Noga Alon**, Tel Aviv University and IAS, Princeton

**"Schramm-MSR Lecture": Random Cayley Graphs**

The study of random Cayley graphs of finite groups is related to the
investigation of Expanders and to problems in Combinatorial Number Theory and
in Information Theory. I will discuss this topic, describing the motivation and
focusing on the question of estimating the chromatic number of a random Cayley
graph of a given group with a prescribed number of generators.

- 11:30
- 12:20
**Fabio Martinelli**, Universita di Roma 3

**Mixing times for constrained spin models**

Consider the following Markov chain on the set of all possible zero-one
labelings of a rooted binary tree of depth L: At each vertex *v*
independently, a proposed new label (equally likely to be 0 or 1) is generated
at rate 1. The proposed update is accepted iff either *v* is a leaf or
both children of *v* are labeled "0”. A natural question is to
determine the mixing time of this chain as a function of L. The above is just
an example of a general class of chains in which the local update of a spin
occurs only in the presence of a special ("facilitating")
configuration at neighboring vertices. Although the i.i.d. Bernoulli
distribution remains a reversible stationary measure, the relaxation to
equilibrium of these chains can be extremely complex, featuring dynamical phase
transitions, metastability, dynamical heterogeneities and universal behavior. I
will report on progress on the mixing times for these models.

- 12:30
- 2:30
**Lunch**, catered, including probability software demos and open problem session - 2:30 -
3:10
**Ronen Eldan**, Microsoft

**A Two-Sided Estimate for the Gaussian Noise Stability
Deficit **

The Gaussian Noise Stability of a set A in Euclidean space is the probability
that for a Gaussian vector X conditioned to be in A, a small Gaussian
perturbation of X will also be in A. Borel's celebrated Isoperimetric
inequality states that a half-space maximizes noise stability among sets with
the same Gaussian measure. We will present a novel short proof of this
inequality, based on stochastic calculus. Moreover, we prove an almost tight,
two-sided, dimension-free robustness estimate for this inequality: We show that
the deficit between the noise stability of a set A and an equally probable
half-space H can be controlled by a function of the distance between the
corresponding centroids. As a consequence, we prove a conjecture by Mossel and
Neeman, who used the total-variation distance.

- 3:10 -
3:30
**Coffee** - 3:30 -
4:10
**Roberto Oliviera**, IMPA

**Mixing of the symmetric exclusion processes in terms of
the corresponding single-particle random walk **

We prove an upper bound for the mixing time of the symmetric exclusion process
on any graph G, with any feasible number of particles. Our estimate is
proportional to the mixing time of the corresponding single-particle random
walk times a log |V| term, where |V| is the number of vertices. This bound
implies new results for symmetric exclusion on expanders, percolation clusters,
the giant component of the Erdos-Renyi random graph and Poisson point
processes. Our technical tools include a variant of Morris's chameleon process.