Proofs of phase transitions by comparison to mean-field theory

Speaker: 

Professor Marek Biskup

Institution: 

UCLA

Time: 

Tuesday, June 6, 2006 - 11:00am

Location: 

MSTB 254

Mean-field theory is one of the most standard tools used by
physicists to analyze phase transitions in realistic systems. However,
regarding rigorous proofs, the link to mean-field theory has been
limited to asymptotic statements which do not yield enough control
of the actual systems. In this talk I will describe a new approach to
this set of problems -- developed jointly with Lincoln Chayes and
Nicolas Crawford -- that overcomes this hurdle in a rather elegant
way. As a conclusion, I will show that a general, ferromagnetic
nearest neighbor spin system on Z^d undergoes a first order phase
transition whenever the mean-field theory indicates one, provided
the dimension d is sufficiently large. Extensions to systems with non
nearest neighbor interactions will also be discussed.

On phase transitions for homopolymers and hetereopolymers.

Speaker: 

Professor Michael Cranston

Institution: 

UCI

Time: 

Tuesday, January 31, 2006 - 11:00pm

Location: 

MSTB 254

Given the measure on random walk paths $P_0$ and a Hamiltonian $H$ the Gibbs perturbation of $H$ defined by
$$\frac{dP_{\beta,t}}{dP_0}=Z^{-1}_{\beta,t}\exp\{\beta H(x)\}$$
with
$$Z_{\beta,t}=\int \exp\{-\beta H(x)\}dP_0(x)$$
gives a new measure on paths $x$ which can be viewed as polymers.
In the case $H(x)=\int_0^t\delta_0(x_{s})ds(\int_0^t\delta_0(x_{s})dW_s)$ we say the resulting measure is concentrated on "homopolymers" ("heteropolymers") and are interested in the influence of dimension and $\beta$ on their behavior.

Phase coexistence of gradient Gibbs measures

Speaker: 

Professor Marek Biskup

Institution: 

UCLA

Time: 

Tuesday, February 21, 2006 - 11:00pm

Location: 

MSTB 254

A gradient Gibbs measure is the projection to the gradient variables $\eta_b=\phi_y-\phi_x$
of the Gibbs measure of the form
$$
P(\textd\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle x,y\rangle}V(\phi_y-\phi_x)\Bigr\}\textd\phi,
$$
where $V$ is a potential, $\beta$ is the inverse temperature and $\textd\phi$ is the product
Lebesgue measure. The simplest example is the (lattice) Gaussian free field
$V(\eta)=\frac12\kappa\eta^2$. A well known result of Funaki and Spohn (and Sheffield)
asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are
characterized by the \emph{tilt}, which is a vector $u\in\R^d$ such that
$E(\eta_b)=u\cdot b$ for any (oriented) edge $b$. I will discuss a simple example
for which this result fails once $V$ is sufficiently non-convex thus showing that
the conditions of Funaki-Spohn's theory are generally optimal. The underlying
mechanism is an order-disorder phase transition known, e.g., from the context
of the $q$-state Potts model with sufficiently large $q$. Based on joint work
with Roman Koteck\'y.

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus.

Speaker: 

Professor Jason Schweinsberg

Institution: 

UCSD

Time: 

Tuesday, February 14, 2006 - 11:00pm

Location: 

MSTB 254

Let x and y be points chosen uniformly at random
from the four-dimensional discrete torus with side length n.
We show that the length of the loop-erased random walk from
x to y is of order n^2 (log n)^{1/6}, resolving a conjecture
of Benjamini and Kozma. We also show that the scaling limit
of the uniform spanning tree on the four-dimensional discrete
torus is the Brownian continuum random tree of Aldous. Our
proofs use the techniques developed by Peres and Revelle,
who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the
d-dimensional discrete torus for d >= 5, in combination with
results of Lawler concerning intersections of
four-dimensional random walks.

Random difference equations with Markov-dependent coefficients

Speaker: 

Professor Alex Roiterstein

Institution: 

University of British Columbia

Time: 

Tuesday, April 18, 2006 - 11:00am

Location: 

MSTB 254

For a class of stationary Markov-dependent sequences
(A_n,B_n)
in R^2, we consider the random linear recursion S_n=A_n+B_n
S_{n-1}, n \in \zz, and show that the distribution tail of its
stationary solution has a power law decay.

Models of polymer dynamics

Speaker: 

Professor Peter March

Institution: 

Ohio State University

Time: 

Tuesday, April 4, 2006 - 11:00am

Location: 

MSTB 254

There is a rich physical literature on polymer dynamics which presents a number of fascinating challenges for mathematicians. We model thermal fluctuation of a polymer in solvent as a curve or loop obeying a stochastic partial differential equation (SPDE). The simplest instance is the so-called Rouse model which is an infinite dimensional Ornstein-Uhlenbeck process satisfying a linear SPDE. We'll review the Rouse model and then describe recent results (a) of Seung Lee on an SPDE for the Rouse model in a half-space with reflecting boundary conditions; and (b) of Scott McKinley on an SPDE model of the hydrodynamic interaction.

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