No abstract

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.

We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,\beta}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,\beta}$ and prove that, for an appropriate centering $c_{N,\beta}$, $NF_{N,\beta}-c_{N,\beta}$ 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

We study the free energy landscape defined by associating to each point in the interior of the ball the free energy corresponding to a thin spherical band around it. This landscape is closely related to several fundamental objects from spin glass theory. For example, the pure states in the decomposition proved by Talagrand (2010) concentrate on bands corresponding to points on the sphere of radius √ N q which asymptotically maximize the free energy, where q it the rightmost point in the support of the overlap distribution. The famous ultrametricity property proved by Panchenko (2013) defines a tree whose branching points have the same property with q < q . We prove that each of those points σ, either a center of a pure state or a branching point, also asymptotically minimizes the (extended) Hamiltonian over the sphere of radius σ .We derive a TAP formula for the free energy for any q in the support of the overlap distribution, expressed by the free energy of an explicit (deterministic) mixture corresponding to the restriction of the Hamiltonian to an appropriate band and a ground state energy. For q = q , the latter free energy has a trivial (replica symmetric) expression. The minimality property also allows us to obtain bounds on the support of the overlap distribution at positive temperature in terms of the overlap distribution of the restriction of the Hamiltonian to the sphere of radius √ N q , in the 0-temperature limit. Those bounds generalize to overlaps of samples from the system at two different temperatures. The latter is used to prove that temperature chaos cannot be detected at the level of free energies, for a class of models arising in the work of Chen and Panchenko (2017).

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.Extremal processes in related models. To put things into context, we now discuss several models of importance, intimately related to the spherical p-spin model. First are the Random Energy Model (REM), in which energy levels are assumed to be independent, and generalized REM (GREM), where correlations are introduced through a tree structure of finite depth. These mathematically tractable models were introduced by Derrida in the 80s [38,39] in order to investigate the phenomenon of replica symmetry breaking exhibited by the Sherrington-Kirkpatrick (SK) model. They have been extensively studied since then and clear connections to spin glass theory have been established, elucidating important concepts in the Parisi theory (see, e.g., the review papers [23] by Bovier and Kurkova and [16] by Bolthausen). Wishing to extend the tree structure mentioned above to account for infinite number of levels one is naturally led to the Branching Brownian Motion (BBM) and Branching Random Walk (BRW), which are of interest on their own. Good sources about the above mentioned models, motivated by connections to spin glass theory and very relevant to the study of extremal processes, are the lecture notes of Bovier [20] and Kistler [48]. Another related model which possesses an implicit hierarchical structure similar to BRW is the 2-Dimensional Discrete Gaussian Free Field (DGFF), see [17].The convergence of the extremal process of the REM model to a PPP of exponential intensity is a classical result of extreme value theory [59,50]. Already for the relatively simple GREM the classical theory is not enough. In the two papers [21,22] Bovier and Kurkova studied the extreme values of the GREM model and a generalization of it, the CREM model. For the GREM model they describe the extremal process in terms of a cascade of PPPs. Their representation implies, in particular, that the process is in general a randomly shifted PPP (SPPP). Convergence in the case of BBM was established independently in two important papers by Arguin, Bovier and Kistler [11] and Aïdékon, Berestycki, Brunet and Shi [4], using somewhat different approaches. The limiting...