arXiv subjects

Damian Pavlyshyn

Publications and source records attributed to Damian Pavlyshyn.

3 recordsLinked to original sources

arXiv2024

Spin glass to paramagnetic transition and triple point in Spherical SK model

This paper studies spin glass to paramagnetic transition in the Spherical Sherrington-Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant $J$ and inverse temperature $β$. The disorder of the system is represented by a general Wigner matrix. We confirm a conjecture of \cite{Baik2016} and \cite{Baik2017}, that the critical window of temperatures for this transition is $β= 1 + bN^{-1/3} \sqrt{\log N}$ with $b\in\mathbb{R}$. The limiting distribution of the scaled free energy is Gaussian for negative $b$ and a weighted linear combination of independent Gaussian and Tracy-Widom components for positive $b$. In the special case where the Wigner matrix is from the Gaussian Orthogonal or Unitary Ensemble, we describe the triple point transition between spin glass, paramagnetic, and ferromagnetic regimes in a critical window for $(β, J)$ around the triple point $(1,1)$: the Tracy-Widom component is replaced by the one parameter family of deformations described by Bloemendal and Virag, \cite{BloVirI}.

math.PR↗

arXiv2022

Comparison of REML methods for the study of phenome-wide genetic variation

It is now well documented that genetic covariance between functionally related traits leads to an uneven distribution of genetic variation across multivariate trait combinations, and possibly a large part of phenotype-space that is inaccessible to evolution. How the size of this nearly-null genetic space translates to the broader phenome level is unknown. High dimensional phenotype data to address these questions are now within reach, however, incorporating these data into genetic analyses remains a challenge. Multi-trait genetic analyses, of more than a handful of traits, are slow and often fail to converge when fit with REML. This makes it challenging to estimate the genetic covariance ($\mathbf{G}$) underlying thousands of traits, let alone study its properties. We present a previously proposed REML algorithm that is feasible for high dimensional genetic studies in the specific setting of a balanced nested half-sib design, common of quantitative genetics. We show that it substantially outperforms other common approaches when the number of traits is large, and we use it to investigate the bias in estimated eigenvalues of $\mathbf{G}$ and the size of the nearly-null genetic subspace. We show that the high-dimensional biases observed are qualitatively similar to those substantiated by asymptotic approximation in a simpler setting of a sample covariance matrix based on i.i.d. vector observation, and that interpreting the estimated size of the nearly-null genetic subspace requires considerable caution in high-dimensional studies of genetic variation. Our results provide the foundation for future research characterizing the asymptotic approximation of estimated genetic eigenvalues, and a statistical null distribution for phenome-wide studies of genetic variation.

stat.AP↗

arXiv2022

An edge CLT for the log determinant of Wigner ensembles

We derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( W_{N}-E_{N}\right)\right\vert,$ where $W_{N}$ is a Wigner matrix, and $E_{N}$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2/3}σ_N$ with $σ_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $σ_N \ll \log^{2} N$. We also extend our CLT to cover spiked Wigner matrices. Our interest in the CLT is motivated by its applications to statistical testing in critically spiked models and to the fluctuations of the free energy in the spherical Sherrington-Kirkpatrick model of statistical physics.

math.PR↗