The effect of disorder on polymer depinning transitions

Speaker: 

Professor Ken Alexander

Institution: 

USC

Time: 

Tuesday, October 30, 2007 - 11:00am

Location: 

MSTB 254

We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.

Limit Theorems and Phase Tranisitions for Homopolymers ,II.

Speaker: 

Professor Michael Cranston

Institution: 

UCI

Time: 

Tuesday, October 2, 2007 - 11:00am

Location: 

MSTB 254

We present a continuation pf work with Molchanov on the behavior of "random walk" oaths under a Gibbs measure which introduces an attraction to the origin with strength depending on a parameter b.
There is a phase transition from a transient or diffusive phase to a globular phase and we discuss behavior at and around the critical value of the parameter .

Spectral Analysis of Brownian Motion with Jump Boundary

Speaker: 

Professor Wenbo Li

Institution: 

University of Delaware

Time: 

Tuesday, August 21, 2007 - 10:00am

Location: 

MSTB 254

Consider a family of probability measures $\{\mu_y : y \in
\partial D\}$ on a bounded open domain $D\subset R^d$ with smooth
boundary.
For any starting point $x \in D$, we run a
a standard $d$-dimensional Brownian motion $B(t) \in R^d $ until it first
exits $D$ at time $\tau$,
at which time it jumps to a point in the domain $D$ according to the
measure $\mu_{B(\tau)}$ at the exit time,
and starts the Brownian motion afresh. The same evolution is repeated
independently each time the process reaches the boundary.
The resulting diffusion process is called Brownian motion with jump
boundary (BMJ).
The spectral gap of non-self-adjoint generator of BMJ, which describes the
exponential
rate of convergence to the invariant measure, is studied.
The main analytic tool is Fourier transforms with only real zeros.

Mean-field dynamics of bosons in a trap: Exchangeability approach

Speaker: 

Professor Marek Biskup

Institution: 

UCLA

Time: 

Thursday, May 31, 2007 - 2:00pm

Location: 

MSTB 254

I will discuss the behavior of N bosons trapped in a potential well subject
to a pairwise interaction. In the mean-field limit -- i.e., when N tends to
infinity while keeping the interaction per particle bounded -- the evolution
of a product state remains, asymptotically, a product state. The single
particle wave-function then evolves according to a non-linear Hartree
equation. Versions of this result have been proved before, e.g., by
Hepp in 1977, Spohn in 1980 or, recently, by Rodnianski and Schlein, but
the proofs are often quite technically involved. I will describe a very simple,
and ideologically correct, proof (for bounded interaction potentials) based
on exchengeability and Stormer's (aka quantum deFinetti) theorem.
Based on recent discussions with Nick Crawford.

Default probabilities, credit derivatives, and computational issues.

Speaker: 

Professor Jean-Pierre Fouque

Institution: 

UCSB

Time: 

Tuesday, May 29, 2007 - 11:00am

Location: 

MSTB 254

The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. We then consider the case of many names and we discuss various ways of creating correlation of defaults. In highly-dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.

Diffusions in random environment and ballistic behavior.We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence.

Speaker: 

Professor Tom Schmitz

Institution: 

UCLA

Time: 

Tuesday, May 15, 2007 - 11:00am

Location: 

MSTB 254

We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average over the environment. One then obtains the so-called annealed measures, that are typically non-Markovian measures.
Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime
('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce
conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix.
As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment.

Large deviations for partition functions of directed polymers and other models in an IID field.

Speaker: 

Professor Iddo Ben-Ari

Institution: 

UCI

Time: 

Tuesday, May 8, 2007 - 11:00am

Location: 

MSTB 254

Consider the partition function of a directed polymer in an IID
field. Under some mild assumptions on the field, it is a well-known fact
that the free energy of the polymer is equal to some deterministic constant
for almost every realization of the
field and that the upper tail large deviations is exponential. In this
talk I'll discuss the lower tail large deviations and present a method
for estimating it. As a consequence, I'll show that the lower
tail large deviations exhibits three regimes, determined by the
tail of the negative part of the field. The method applies to other
oriented models and can be adapted to non-oriented models as well. This
work extends the results of a recent paper by Cranston Gautier and
Mountford. A preprint is availabe on www.math.uci.edu/~ibenari

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