UC San Diego Probability Seminar (2021-2022)

The seminar meets on Thursdays via Zoom from 11 AM to 12 PM PT unless noted otherwise. Please send me an email if you would like to be added to the mailing list or if you have a suggestion for a speaker. Invite links to talks will be sent to the mailing list.

Fall

October 7: Brett Kolesnik (UC San Diego)

Abstract: A graph \(G\) is said to \(H\)–percolate if, by iteratively completing copies of \(H\) minus an edge, all missing edges are eventually added. This process was introduced in early work of Bollobás (1967), and studied more recently by Balogh, Bollobás and Morris (2012) in the case that \(G\) is the Erdős–Rényi random graph \(G(n,p)\). In this talk, we will discuss our recent joint work with Zsolt Bartha (TU Eindhoven), which locates the critical threshold \(p_c\) (at which point \(G(n,p)\) is likely to \(H\)–percolate) for all "reasonably balanced" graphs \(H\). When \(H\) is the complete graph \(K_r\), this answers an open problem of Balogh, Bollobás and Morris.



October 14: Wolfgang König (Weierstrass Institute Berlin (WIAS) and TU Berlin)

Abstract: We study an inhomogeneous sparse random graph, \(G_N\), on \([N] = \{1,\dots,N\}\) as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit of large \(N\), we consider the sparse regime, where the average degree is \(O(1)\). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size \(\asymp N\)). In doing so, we derive explicit logarithmic asymptotics for the probability that \(G_N\) is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of \(G_N\). In particular, we recover the criterion for the existence of the phase transition of the emergence of a giant cluster given in [BJR07].
Joint work with Luisa Andreis (Florence), Heide Langhammer (WIAS) and Robert Patterson (WIAS).



October 28: Gaultier Lambert (University of Zurich)

Abstract: TBA



November 4: Evgeni Dimitrov (Columbia)

Abstract: Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability. Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others. Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.



November 18: Johannes Alt (University of Geneva and Courant Institute)

Abstract: TBA



December 2: Nike Sun (MIT)

Abstract: TBA



* There will be no seminar on November 11 (Veterans Day) and November 25 (Thanksgiving)