## EVENT DETAILS AND ABSTRACT

**Probability Seminar**
**Title:** A Solvable Homopolymer Model

**Speaker:** Mike Cranston

**Speaker Info:** University of California, Irvine

**Brief Description:**

**Special Note**:

**Abstract:**

With the reference measure ${P}^x$ induced by Brownian motion on ${\bf{R^3}}$ starting at $0$ we wish to define,
for $\db \in{\bf{R}},$ the Gibbs probability measure $\overline{P}^x_{\db,t}$ with density with respect to $P^0$ heuristically given by
\begin{eqnarray}
\frac{d\overline{P}^x_{\db,t}}{d \,P^x}=Z_{\db,t}^{-1}e^{\db \int_0^t\delta_0(\omega_s)ds}
\end{eqnarray}
where $Z_{\db,t}\equiv E^x[e^{\db \int_0^t\delta_0(\omega_s)ds}].$ We achieve this using a one parameter family of self-adjoint extensions of the Laplacian on $C_c({\bf{R^3-\{0\}}}).$ This Gibbs probability measure provides a simple continuum model for a homopolymer with an attractive potential at the origin. In this talk we give a comprehensive study of the behavior of paths with respect to these Gibbs measures. In particular, there is a phase transition in the behavior of these paths from diffusive behavior for $\db\le0$ to positive recurrent behavior for $\db>0.$ The critical value is determined by means of the spectral properties of the operator $\mathcal{H}_\db,$ the self-adjoint extension of the Laplacian on $C_c({\bf{R^3-\{0\}}})$ corresponding to the parameter value $\db.$ This corresponds to a transition from a diffusive or stretched out phase to a globular phase for the polymer. We consider various quantities associated to the paths and examine their behavior near the critical point. We also draw comparisons to a similar model on ${\bf{Z^3}}$ previously studied by the authors. The program can also be carried out for stable processes in low dimensions. This talk is based on joint work with L. Koralov, S. Molchanov, N. Squartini and B. Vainberg.

**Date:** Tuesday, November 01, 2016

**Time:** 3:00PM

**Where:** Lunt 105

**Contact Person:** Elton Hsu

**Contact email:** ehsu@math.northwestern.edu

**Contact Phone:** 1-8541

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