arXiv subjects

Han Gia Le

Publications and source records attributed to Han Gia Le.

4 recordsLinked to original sources

arXiv2025

Free energy fluctuations in SK and related spin glass models: A literature survey

Over the past 50 years, spin glass models have generated a broad range of literature in mathematics, physics, and computer science. There has been much progress in characterizing and proving the limiting free energy of various models, stemming from the original formulas of Parisi. Comparatively less is known about the more detailed topic of free energy fluctuations. This paper concerns a family of models in which there has been considerable progress on fluctuations, namely the Sherrington-Kirkpatrick (SK) and spherical Sherrington-Kirkpatrick (SSK) models, along with their multi-species analogs. We present a survey of the literature on free energy fluctuations in these 2-spin models, discussing results from different temperature regimes, with and without an external field, including results on phase transitions.

math.PR↗

arXiv2025

Order of fluctuations of the free energy in the positive semi-definite MSK model at critical temperature

In this note, we consider the multi-species Sherrington-Kirkpatrick spin glass model at its conjectured critical temperature, and we show that, when the variance profile matrix $Δ^2$ is positive semi-definite, the variance of the free energy is $O(\log^2N)$. Furthermore, when one approaches this temperature threshold from the low temperature side at a rate of $O(N^{-α})$ with $α>0$, the variance is $O(\log^2N+N^{1-α})$. This result is a direct extension of the work of Chen and Lam (2019) who proved an analogous result for the SK model, and our proof methods are adapted from theirs.

math.PR↗

arXiv2023

Free energy of the bipartite spherical SK model at critical temperature

The spherical Sherrington-Kirkpatrick (SSK) model and its bipartite analog both exhibit the phenomenon that their free energy fluctuations are asymptotically Gaussian at high temperature but asymptotically Tracy-Widom at low temperature. This was proved in two papers by Baik and Lee, for all non-critical temperatures. The case of critical temperature was recently computed for the SSK model in two separate papers, one by Landon and the other by Johnstone, Klochkov, Onatski, Pavlyshyn. In the current paper, we derive the critical temperature result for the bipartite SSK model. In particular, we find that the free energy fluctuations exhibit a transition when the temperature is in a window of size $n^{-1/3}\sqrt{\log n}$ around the critical temperature, the same window for the SSK model. Within this transitional window, the asymptotic fluctuations of the free energy are the sum of independent Gaussian and Tracy-Widom random variables.

math.PR↗

arXiv2023

An edge CLT for the log determinant of Laguerre beta ensembles

We obtain a CLT for $\log|\det(M_n-s_n)|$ where $M_n$ is a scaled Laguerre $β$ ensemble and $s_n=d_++σ_n n^{-2/3}$ with $d_+$ denoting the upper edge of the limiting spectrum of $M_n$ and $σ_n$ a slowly growing function ($\log\log^2 n\llσ_n\ll\log^2 n$). In the special cases of LUE and LOE, we prove that the CLT also holds for $σ_n$ of constant order. A similar result was proved for Wigner matrices by Johnstone, Klochkov, Onatski, and Pavlyshyn. Obtaining this type of CLT of Laguerre matrices is of interest for statistical testing of critically spiked sample covariance matrices as well as free energy of bipartite spherical spin glasses at critical temperature.

math.PR↗