## Departmental PhD Thesis Exam – Jackson Feng

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, August 12, 2015
11:10 a.m.
BA6183

PhD Candidate: Jackson Feng
Supervisor:  Jeremy Quastel
Thesis title: Rescaled Directed Random Polymer in Random Environment in Dimension 1 + 2

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Abstract:

Directed random polymer in random environment is a statistical physics model. Heuristically, it concerns a long, directed chain of molecules evolving in some random background, and one of the questions one asks is how is long term behaviour of the polymer affected by strength of the noise, which in the model is controlled by a parameter $\beta$ called inverse temperature.

In this thesis, we will let $\beta$ depend on length of the polymer and consider the model in time-space dimension $1+2$. Also, we will work with continuous time and space, and white noise random environment.

The main result is if we rescale $\beta$ as $\beta_t=\frac{\alpha_t}{\sqrt{\log t}}$, where $\alpha^2_t=4\pi + \frac{\sigma}{\log t}$ for $\sigma\in \mathbb{R}$, then we observe non-Gaussian fluctuation of the random polymer.

Along the way, we give a formula for the exponential moment of the local time of two dimensional Brownian motion starting at $0$, and show the two dimensional delta Bose gases are correlated.

A copy of the thesis can be found here: Feng_Zisheng_201606_PhD_thesis