# Spring 2019

January 10: TBA (TBA)

Title: TBA

Abstract: TBA

January 17: Boris Pittel (The Ohio State University)

Title: On the Bollobás-Riordan random pairing model of preferential attachment graph

Abstract: The Bollobás-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied.  The degrees of the first $n^{\frac{m}{m+2} - \epsilon}$ vertices are jointly, uniformly asymptotic to an explicit function of the inter-arrival times of a standard Poisson process.  Further it is shown that all these vertices have degree $n^{\frac{\epsilon (m+2)}{m}}$ with probability tending to $1$ as $n$ goes to infinity.  We bound the probability that there exists a pair of large vertex sets with no edges joining them, and use it to identify the ranges of vertex sets that are exponentially unlikely to be isolated, or likely to be vertex-expanding.

January 24:

Title: TBA

Abstract: TBA

January 31: TBA (TBA)

Title: TBA

Abstract: Alex Furman to give the Colloquium: https://web.math.osu.edu/colloquium/.  We may wish to leave this day open.

February 7: Elizabeth Meckes (Case Western Reserve University)

Title: On the eigenvalues of Brownian motion on U(n)

Abstract:  For U_t a standard Brownian motion on U(n), we consider the evolution of the corresponding eigenvalue distributions.  More specifically, we consider the induced stochastic process of empirical spectral measures, and give uniform quantitative almost-sure estimates over fixed time intervals of the distance between these spectral measures and the corresponding measures in a deterministic parametrized family nu_t of large-n limiting measures.

This is joint work with Tai Melcher.

February 14:  Murong Xu   (The Ohio State University)

http://math.osu.edu/events/combinatorics-seminar-murong-xu

February 21: Shai Evra (IAS)

https://math.osu.edu/events/combinatorics-seminar-shai-evra

February 28: (Not available)

Title: TBA

Abstract: TBA

March 7: Henry Towsner  (University of Pennsylvania)

Title: Nonstandard analysis and new standard proof of the containers theorem

Abstract: The hypergraph containers theorem, introduced by Balogh-Morris-Samotij and Saxton-Thomason, is a breakthrough in extremal combinatorics which gives a tool for controlling the number of independent sets in finite hypergraphs. (We do not assume any familiarity with the containers method!) The original proof depends on giving a program which constructs ‘containers’ for independent sets and analyzing its behavior to ensure the construction has the right properties. Inspired by the idea of pseudofinite dimension in ultraproducts (with which familiarity is also not assumed), we give a new, elementary and nonalgorithmic, proof of the containers theorem which gives an direct characterization of the ‘containers’.

Joint work with Bernshteyn, Delcourt, and Tserunyan.

Abstract: TBA

March 14: Spring break (Beach University of South Florida)

Title: Spring break

Abstract: Spring break

March 21: Thai-Hoang Le (University of Mississippi)

https://math.osu.edu/events/combinatorics-seminar-thai-hoang-le

March 28: Robert Hough (Stony Brook)

https://math.osu.edu/events/combinatorics-seminar-robert-hough

April 4: Jack Hanson (CUNY)

Title: Universality of the time constant for critical first-passage percolation on the triangular lattice

Abstract: We consider first-passage percolation (FPP) on the triangular lattice with vertex weights whose common distribution function F satisfies F(0) = 1/2. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T_n the first-passage time from 0 to the boundary of the box of sidelength n, we show existence of the time constant – the limit of T_n / log n – and find its exact value to be I / (2 (√ 3 π). (Here I = inf{x > 0 : F(x) > 1/2}.) This shows that the time constant is universal, in the sense that it is insensitive to most details of F. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of I, under the optimal moment condition on F.

April 11: Hanbaek Lyu (UCLA)

Title: Phase transition in random contingency tables with non-uniform margins

Abstract: Contingency tables are matrices with nonnegative integer entries with fixed row and column margins. Understanding the structure of uniformly chosen contingency table with given margins is an important problem especially in statistics. For parameters $n,\delta,B,$ and $C$, let $X=(X_{k\ell})$ be the random uniform contingency table whose first $\lfloor n^{\delta} \rfloor$ rows and columns have margin $\lfloor BCn \rfloor$ and the other $n$ rows and columns have margin $\lfloor Cn \rfloor$. For any $0<\delta<1$, we establish a sharp phase transition of the limiting distribution of each entries of $X$ at the critical value $B_{c}=1+\sqrt{1+1/C}$. One of our main result shows that, for $1/2<\delta<1$, all entries have uniformly bounded expectation for $B<B_{c}$, but the mass concentrates at the smallest block and grows in the order of $n^{1-\delta}$ for $B>B_{c}$. We also establish a strong law of large numbers for the row sums within blocks. Joint work with Igor Pak and Sam Dittmer.

April 18: James D. Cordeiro (University of Dayton)

Title: The Role of the Group Inverse in the Ergodicity of Level-Dependent Quasi-Birth-and-Death Processes (LDQBDs)

Abstract: Quasi-birth-and-death (QBD) processes are a class of structured Markov chains that extend the classical Birth-Death model by permitting state transitions that may occur between births and deaths. Its level-dependent generalization, the LDQBD, has generated a considerable amount of interest due to the fact that a large number of queueing models belong to this class of processes, and yet an analytic steady-state criterion has not been developed up to the present time. In this presentation, we describe the application of Foster-Lyapunov drift to the determination of necessary and sufficient analytic stability criteria for a subclass of discrete-time LDQBD processes whose transition matrices converge over block rows. Particular emphasis is placed on the role of Markov generalized inverse theory in satisfying the requirements of this drift condition. This is the first known application of the Markov group inverse to an infinite-state process via application to levels of the transition probability matrix.

April 25: Benjamin Landon (MIT)

https://math.osu.edu/events/combinatorics-seminar-benjamin-landon