Probability Day

Speaker : Atul Shekhar (TIFR CAM)

Abstract : For a random polynomial with iid roots in a disc, we consider its level sets (aka lemniscates). We obtain asymptotics for the expected number of connected components. This number shows a distinct behaviour at the critical radius R = 1. This can be accredited to a pairing phenomenon between roots and critical points of the given polynomial. This talk is based on a joint work with Koushik Ramachandran andSubhajit Ghosh.

Speaker : D Yogeshwaran (ISI Bangalore)


Abstract : This talk will focus on the asymptotic variances of random point clouds under invariant transports. The emphasis will be on the phenomenon of hyperuniformity, characterized by vanishing asymptotic variance, which reflects long-range order and regularity in point clouds. We will see examples and non-examples of hyperuniform point clouds as well as their realizability by ‘good’ transports or matchings to a lattice. We discuss some sufficient conditions on transport maps that preserve asymptotic variance. The conditions will be illustrated by proving or refuting hyperuniformity in examples.

Speaker : Sudeshna Bhattacharjee (IISc)

Abstract : Weak convergence of the centered and scaled sequences of largest eigenvalues of n × n GUE matrices, and of the lengths of the longest increasing subsequences of uniformly random permutations of length n, to the GUE Tracy-Widom distribution, are two primary examples of the KPZ fluctuations. In this talk we address a question posed by Kalai: what is the analogue of the classical law of iterated logarithm for these sequences? In particular, we show that after a further scaling of (log n) 2/3 (resp. ((log n)1/3), the lim sup (resp. lim inf) of the centered and scaled sequences almost surely converge to some non-zero and finite constants, which we explicitly determine. This answers a question left open by Paquette and Zeitouni. One key ingredient in solving these problems was understanding the scale at which decorrelation occurs in these
sequences. For this we use the connection with the Brownian and Poissonian last passage percolation models and use the geometric inputs from these models. This is based on a joint work with Baslingker, Basu, and Krishnapur (arXiv:2410.11836).

Speaker : Arvind Ayyer (IISc)

Abstract : The Tsetlin library is a random-to-top shuffle process on permutations. A q-deformation using the Hecke algebra can also be implemented on complete flags over the finite field Fnq. The semigroup underlying the random-to-top Markov chain on flags is a left regular band, which provides a method to compute the stationary distribution and eigenvalues. We consider a further generalisation by introducing inhomogeneous rates. This is work in collaboration with Sarah Brauner, Jan de Gier and Anne Schilling.

Speaker: Nathana ̈el Berestycki (University of Vienna)

Abstract: Models of statistical mechanics are notoriously hard to analyse at their critical point. In the 1980s, physicists came up with a very novel and far-reaching approach to such problems in two dimensions. This involves analysing them on random surfaces such as random triangulations instead of Euclidean lattices, in combination with a remarkable formula (known as KPZ, after Knizhnik, Polyakov and Zamolodchikov) which yields a precise correspondence between the geometric properties of the models in their random geometry formulation and their Euclidean analogues. I will give an introduction to some of these ideas and illustrate them with a novel example, the fully packed loop O(n) model on random triangulations, which is also bijectively equivalent to the self-dual Fortuin-Kasteleyn percolation model on planar maps (joint work with William Da Silva). Time-permitting, I will also explain how one can give a rigorous derivation of the KPZ formula in the closely related context of Liouville quantum gravity.

Speaker: Ratul Biswas (NSF–Simons National Institute for Theory and Mathematics in Biology)

Abstract: In this talk, I shall present a unifying approach to study the replica-symmetric regime of general dilute spin glasses. In particular, we identify two regimes high-temperature and subcritical dilution - where we observe a large class of popular spin glasses exhibiting replica-symmetric behavior. Our approach reproves several well-known formulas for the free and ground state energies of some of these models and derives new ones for some others. Joint work with Arnab Sen and Wei-Kuo Chen.

Speaker: Aditya Thorat (TIFR CAM–ICTS)

Abstract: Finitely dependent processes are a natural generalisation of independent processes. We will discuss results which try to answer the following question: Given a combinatorial models on Zd specified by local constraints (e.g. proper k colorings, hard core model, dimer model), does there exists a stationary finitely dependent processes supported on the model?

Speaker: Shaibal Karmakar (TIFR CAM–ICTS)

Abstract: This talk concerns upper tail large deviations for subgraph counts in sparse Erd ̋os-R ́enyi graphs. I will present recent results showing that for irregular subgraphs, sharp large deviation rates can be obtained well into the sparse regime, and that the dominant contribution comes from localized structures rather than global edge density shifts. Based on joint work with Anirban Basak, the talk will highlight phase transitions between Poisson-type behaviour and localization, as well as connections to mean-field variational problems.