Estimation in spin glasses: A first step

The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of neural networks and the Ising spin glass are all models of binary data belonging to the one-parameter exponential family with quadratic sufficient statistic. Under bare minimal conditions, we establish the $\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural parameter in this family, even at critical temperatures. Since very little is known about the low and critical temperature regimes of these extremely difficult models, the proof requires several new ideas. The author's version of Stein's method is a particularly useful tool. We aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions.Comment: Published in at http://dx.doi.org/10.1214/009053607000000109 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Canonical decomposition of operators associated with the symmetrized polydisc

A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $\Gamma_n$ is a spectral set is called a $\Gamma_n$-contraction. We show that every $\Gamma_n$-contraction admits a decomposition into a $\Gamma_n$-unitary and a completely non-unitary $\Gamma_n$-contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $\Gamma_n$ and $\Gamma_n$-contractions.Comment: Complex Analysis and Operator Theory, Published online on August 28, 2017. arXiv admin note: text overlap with arXiv:1610.0093

Absence of replica symmetry breaking in the random field Ising model

It is shown that replica symmetry is not broken in the random field Ising model in any dimension, at any temperature and field strength, except possibly at a measure-zero set of exceptional temperatures and field strengths.Comment: 11 pages. To appear in Commun. Math. Phy

Avatars of Margulis invariants and proper actions

In this article, we interpret affine Anosov representations of any word hyperbolic group in $\mathsf{SO}_0(n-1,n)\ltimes\mathbb{R}^{2n-1}$ as infinitesimal versions of representations of word hyperbolic groups in $\mathsf{SO}_0(n,n)$ which are both Anosov in $\mathsf{SO}_0(n,n)$ with respect to the stabilizer of an oriented $(n-1)$-dimensional isotropic plane and Anosov in $\mathsf{SL}(2n,\mathbb{R})$ with respect to the stabilizer of an oriented $n$-dimensional plane. Moreover, we show that representations of word hyperbolic groups in $\mathsf{SO}_0(n,n)$ which are Anosov in $\mathsf{SO}_0(n,n)$ with respect to the stabilizer of an oriented $(n-1)$-dimensional isotropic plane, are Anosov in $\mathsf{SL}(2n,\mathbb{R})$ with respect to the stabilizer of an oriented $n$-dimensional plane if and only if its action on $\mathsf{SO}_0(n,n)/\mathsf{SO}_0(n-1,n)$ is proper. In the process, we also provide various different interpretations of the Margulis invariant.Comment: 40 page