# The Polaron Measure

## Srinivasa Varadhan (NYU Courant)

### Thursday, September 13, 2018 4:00 pmSCHN 151 (Storrs)

The Polaron measure on $[-T,T]$ is defined as $Q^\alpha_T$ given by its density with respect to the process $P$ of three dimensional Brownian increments:

$$\frac{d Q^\alpha_T}{d P} = \frac{1}{Z(\alpha,T)} \exp \left[ \frac{\alpha}{2} \int_{-T}^T \int_{-T}^T \frac{e^{-|t-s|}}{|x(t)-x(s)|} dt ds\right]$$

$Z(\alpha,T)$ is the normalizing constant. In the eighties a conjecture of Pekar that claims that

$$\lim\limits_{\alpha \to \infty} \frac{1}{\alpha^2}\lim_{T\to\infty} \frac{1}{2T} \log Z (\alpha,T) = g_0$$

where

$$g_0=\sup\limits_{\|\phi\|_2=1} \left[ \int_{R^3} \int_{R^3} \frac{\phi^2(x) \phi^2(y)}{|x-y|} dx dy - \frac 12 \int | \nabla \phi|^2(x) dx\right]$$

was proved. We now investigate the behavior of the measure $Q^\alpha_T$ as $T\to\infty$ followed by $\alpha \to \infty$.

Comments: Distinguished Lecture for the special semester in probability