## Departmental PhD Thesis Exam – Andrew Stewart

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

Tuesday, June 28, 2016
3:10 p.m.
BA6183

PhD Candidate:  Andrew Stewart
Co-Supervisors:  Balint Virag
Thesis title:  On the scaling limit of the range of a random walk bridge on regular trees

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Abstract:

Consider a nearest neighbour random walk $X_n$ on the $d$-regular tree ${\Bbb T}_d$, where $d\geq 2$, conditioned on $X_n = X_0$. This is known as the random walk bridge. We derive Gaussian-like tail bounds for the return probabilities of the random walk bridge on the scale of $n^{1/2}$. This contrasts with the case of the unconditioned random walk, where Gaussian-like tail bounds exists on the scale of $n$.

We introduce the notion of the infinite bridge, which is known to arise as the distributional limit of the random walk bridge. We also establish some preliminary facts about the infinite bridge.

By showing that the Brownian Continuum Random Tree (BCRT) is characterized by its random self-similarity property, we prove that the range of the random walk bridge converges in distribution to the BCRT when rescaled by $Cn^{-1/2}$ for an appropriate constant $C$

A draft of the thesis can be found here: andrew-stewart-thesis