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# body{ background-color: #e6ebef } Fall Semester 2023 Student Probability Seminar

Welcome to the Columbia Student Probability Seminar. In this seminar, we will study the paper Directed Landscape written by Duncan Dauvergne, Janosch Ortmann, and Balint Virag. Here is a link to the paper. Our talks will be held in room Mudd 627 on Wednesday from 5:25 PM to 6:25PM.

## Schedule of Talks

Date Speaker content
09/20/2023 Jiyue Zeng Introductions of Last Passage Time, Airy Line Ensembles, Airy process, melon map, Brownian Last Passage Percolations.
09/27/2023 Jiyue Zeng We discuss section three of the paper which covers the basic geometric properties of last passage time and last passage path.
10/11/2023 Victor Hugo Daniel We discuss section four of the paper which proves that the last passage percolation over a set of continuous functions is the same as the last passage percolation over the melon maps, i.e., f[(x,n) -> (y,1)] = Wf[(x,n) -> (y,1)].
10/18/2023 Hindy Drillick We discuss section five of the paper which covers the important result that last passage percolation across the bottom n-k lines can be translated to the backward first passage percolation over the top k lines.
10/25/2023 Jiyue Zeng We discuss the first part of chapter six of the paper. We will discuss Airy Line Ensembles as the scaling limit of last passage percolations over Brownian melons and we characterize the mean and fluctuations of LPP in this form, WB[(x,n) -> (z,k)].
11/01/2023 Jiyue Zeng We discuss the second part of chapter six of the paper. We will use the theorems proved last time to characterize the mean and fluctuations of LPP over Airy Line Ensembles, i.e., A[(x,n) -> (z,k)].
11/08/2023 Zoe Himwich We discuss the first lemma from chapter 7. It shows the concentration of jump time from K+1st line to Kth line in last passage percolation over Brownian melon.
11/15/2023 Zoe Himwich We discuss the second lemma from chapter 7. It shows that under the condition that initial starting locations are close enough, two brownian last passage paths will almost surely intersect.
11/29/2023 Victor Hugo Daniel We discuss chapter 8. We will prove uniqueness in law and existence of Airy sheet. We start by proving tightness.
12/06/2023 Victor Hugo Daniel We continue the discussion last time on constructing Airy sheet.