arXiv subjects

Eliran Subag

Publications and source records attributed to Eliran Subag.

28 records · Page 2Linked to original sources

arXiv2019

The generalized TAP free energy II

In a recent paper [14], we developed the generalized TAP approach for mixed $p$-spin models with Ising spins at positive temperature. Here we extend these results in two directions. We find a simplified representation for the energy of the generalized TAP states in terms of the Parisi measure of the model and, in particular, show that the energy of all states at a given distance from the origin is the same. Furthermore, we prove the analogues of the positive temperature results at zero temperature, which concern the ground-state energy and the organization of ground-state configurations in space.

math.PR↗

arXiv2018

Geometry and temperature chaos in mixed spherical spin glasses at low temperature - the perturbative regime

We study the Gibbs measure of mixed spherical $p$-spin glass models at low temperature, in (part of) the 1-RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius $\sqrt N q_\star$, where $q_\star^2$ is the rightmost point in the support of the overlap distribution. We also show that the relevant critical points are pairwise orthogonal for two different low temperatures. This allows us to explain why temperature chaos occurs for those models, in contrast to the pure spherical models.

math.PR↗

arXiv2016

The complexity of spherical p-spin models - a second moment approach

Recently, Auffinger, Ben Arous, and Černý initiated the study of critical points of the Hamiltonian in the spherical pure $p$-spin spin glass model, and established connections between those and several notions from the physics literature. Denoting the number of critical values less than $Nu$ by $\mbox{Crt}_{N}(u)$, they computed the asymptotics of $\frac{1}{N}\log(\mathbb{E}\mbox{Crt}_{N}(u))$, as $N$, the dimension of the sphere, goes to $\infty$. We compute the asymptotics of the corresponding second moment and show that, for $p\geq3$ and sufficiently negative $u$, it matches the first moment: \[ \mathbb{E}\left\{ \left(\mbox{Crt}_{N}\left(u\right)\right)^{2}\right\} /\left(\vphantom{\left(\mbox{Crt}_{N}\left(u\right)\right)^{2}}\mathbb{E}\left\{ \mbox{Crt}_{N}\left(u\right)\right\} \right)^{2}\to1. \] As an immediate consequence we obtain that $\mbox{Crt}_{N}(u)/\mathbb{E}\{ \mbox{Crt}_{N}(u)\} \to 1$, in $L^2$ and thus in probability. For any $u$ for which $\mathbb{E}\mbox{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments match on an exponential scale.

math.PR↗

arXiv2016

The extremal process of critical points of the pure $p$-spin spherical spin glass model

Recently, sharp results concerning the critical points of the Hamiltonian of the $p$-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points - that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM.

math.PR↗

arXiv2016

The geometry of the Gibbs measure of pure spherical spin glasses

We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,β}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,β}$ and prove that, for an appropriate centering $c_{N,β}$, $NF_{N,β}-c_{N,β}$ is a tight sequence. We establish the absence of temperature chaos and analyze the transition rate to disorder chaos of the Gibbs measure and ground state. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical `bands' centered at deep minima, playing the role of so-called `pure states'. For the pure models, the latter makes precise the so-called picture of `many valleys separated by high mountains' and significant parts of the TAP analysis from the physics literature.

math.PR↗

arXiv2015

Random geometric complexes in the thermodynamic regime

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called `thermodynamic' regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer-Vietoris exact sequences. The Mayer-Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.

math.PR↗

arXiv2014

Freezing and decorated Poisson point processes

The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy $L[θ_y f]=g(y-τ_f)$ for any nonzero, nonnegative, compactly supported, continuous function $f$, where $θ_y$ is the shift operator, $τ_f$ is a real number that depends on $f$, and $g$ is a real function that is independent of $f$. We show that, under some assumptions, this property characterizes the structure of SDPPP. Moreover, when it holds, we show that $g$ has to be a convolution of the Gumbel distribution with some measure. The above property of the Laplace functional is closely related to a `freezing phenomenon' that is expected by physicists to occur in a wide class of log-correlated fields, and which has played an important role in the analysis of various models. Our results shed light on this intriguing phenomenon and provide a natural tool for proving an SDPPP structure in these and other models.

math.PR↗

arXiv2013

Rotation and scale space random fields and the Gaussian kinematic formula

We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI (functional magnetic resonance imaging) brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data. The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper. This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley-Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations. Previously only the two-dimensional case could be covered, and then only via computer algebra. By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer.

math.PR↗

arXiv2013

A Lower Bound for the Mixing Time of the Random-to-Random Insertions Shuffle

The best known lower and upper bounds on the mixing time for the random-to-random insertions shuffle are $(1/2-o(1))n\log n$ and $(2+o(1))n\log n$. A long standing open problem is to prove that the mixing time exhibits a cutoff. In particular, Diaconis conjectured that the cutoff occurs at $3/4n\log n$. Our main result is a lower bound of $t_n = (3/4-o(1))n\log n$, corresponding to this conjecture. Our method is based on analysis of the positions of cards yet-to-be-removed. We show that for large $n$ and $t_n$ as above, there exists $f(n)=Θ(\sqrt{n\log n})$ such that, with high probability, under both the measure induced by the shuffle and the stationary measure, the number of cards within a certain distance from their initial position is $f(n)$ plus a lower order term. However, under the induced measure, this lower order term is strongly influenced by the number of cards yet-to-be-removed, and is of higher order than for the stationary measure.

math.PR↗

arXiv2010

Persistent Homology for Random Fields and Complexes

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.

math.PR↗