# Columbia/Courant joint probability seminar, Fall 2018

## November 30th, 2018 ; WWH 512 (Courant Institute)

• 9:30-10:30am  Konstantin Matetski (Columbia University) A class of models, converging to the KPZ fixed point.

The KPZ universality class contains random growth models, directed random polymers, stochastic Hamilton-Jacobi equations. It is characterized by unusual scale of fluctuations, some of which appeared earlier in random matrix theory, and which depend on the initial data. The scaling invariant Markov process at the center of the KPZ universality class, the KPZ fixed point, has been obtained by Matetski, Quastel and Remenik, by solving one model in the class, TASEP, and passing to the limit. In the talk a broader class of models will be described, for which solutions can be obtained in a similar way. This class includes such knows models as PushASEP, discrete time TASEP with sequential and parallel updates. This class is closed under composition of models, which in particular allows to obtain a solution of TASEP with stationary initial data. (Joint work with J. Quastel and D. Remenik).

• 10:30-11am

Coffee break

• 11am-12pm Fan Yang (UCLA) Delocalization for Random Band Matrices in Dimension 2 and Higher.

We consider Hermitian random band matrices $H$ in dimension $d$, where the entries $h_{xy}$, indexed by $x,y \in [1,N]^d$, vanishes if $|x-y|$ exceeds the band width $W$. It is conjectured that a sharp transition of the eigenvalue and eigenvector statistics occurs at a critical band width $W_c$, with $W_c=\sqrt{N}$ in $d=1$, $W_c=\sqrt{\log N}$ in $d=2$, and $W_c=O(1)$ in $d\ge 3$.
Recently, Bourgade, Yau and Yin proved the bulk universality and eigenvector delocalization for 1D random band matrices with generally distributed entries and band width $W\gg N^{3/4}$. The same method applied to higher dimensions will imply eigenvector delocalization under $W\gg N^{1/2}$. We will show that a weaker delocalization in certain averaged sense holds under a weaker assumption $W\gg N^{2/(2+d)}$. The proof depends crucially on a delicate fluctuations averaging estimate for the Green's function of $H$. This improves the previous result $W\gg N^{(d+1)/(2d+1)}$ by He and Marcozzi and $W\gg N^{6/(d+6)}$ by Erdos and Knowles. Based on Joint work with Jun Yin.

• 12-1pm Richard Kenyon (Brown University) Unicycles in $\mathbb{Z}^2$ and $\mathbb{Z}^3$

A (spanning) unicycle, or cycle-rooted tree, is a spanning subgraph with exactly one cycle: a spanning tree to which an edge has been added. We study uniform unicycles in $\mathbb{Z}^2$ and $\mathbb{Z}^3$, using a noncommutative generalization of the matrix-tree theorem. This is joint work with David Wilson.