Unlike most books reviewed in the Intelligencer this is definitely a textbook. It assumes knowledge one might acquire in the first two years of an undergraduate mathematics program – basic mathematical probability, plus linear algebra, a little graph theory and the infamous concept of “mathematical maturity”. It has the theorem-proof style of pure mathematics, but with friendly explanations of intuition and motivation.

Markov Chains and Mixing Times: Second Edition

Limit Theorems of Probability Theory.

I Introduction.- II Galton-Watson trees.- III Branching random walks and martingales.- IV The spinal decomposition theorem.- V Applications of the spinal decomposition theorem.- VI Branching random walks with selection.- VII Biased random walks on Galton-Watson trees.- A Sums of i.i.d. random variables.- References.

http://www.proba.jussieu.fr/pageperso/zhan/pdffile/brw.pdf

Branching random walks

In this expository article, we survey the rapidly emerging area of random 
geometric simplicial complexes. Random geometric complexes may be viewed as higher-dimensional generalizations of random geometric graphs, where vertices are generated by a random point process, 
and edges are placed based on proximity. Extending the notion of connected components and cycles in graphs, the main object of study has been the homology of these 
complexes. We review the results known to date about the probabilistic behavior of the homology (and related structures) generated by these random complexes.

Topology of random geometric complexes: a survey

In this expository article, we survey the rapidly emerging area of random geometric simplicial complexes.

Recently, Auffinger, Ben Arous and Cerný 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 $\operatorname{Crt}_{N}(u)$, 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: \[\mathbb{E}\{(\operatorname{Crt}_{N}(u))^{2}\}/(\mathbb{E} \{\operatorname{Crt}_{N}(u)\})^{2}\to1.\] As an immediate consequence we obtain that $\operatorname{Crt}_{N}(u)/\mathbb{E}\{\operatorname{Crt}_{N}(u)\}\to1$, in $L^{2}$, and thus in probability. For any $u$ for which $\mathbb{E}\operatorname{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments match on an exponential scale.

/pdf/the-complexity-of-spherical-p-spin-models-a-second-moment-vlv23tecgd.pdf

The complexity of spherical $p$-spin models—A second moment approach

We analyze the statics for pure p-spin spherical spin glass models with $$p\ge 3$$
 , at low enough temperature. With $$F_{N,\beta }$$
 denoting the free energy, we compute the (logarithmic) second order 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 further establish the absence of temperature chaos. 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 ‘pure states’. For the pure models, the latter makes precise the picture of ‘many valleys separated by high mountains’ and significant parts of the TAP analysis from the physics literature.

The geometry of the Gibbs measure of pure spherical spin glasses

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.

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.

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 $${L\left[\theta_{y}f\right]=g\left(y-\tau_{f}\right)}$$
 for any nonzero, nonnegative, compactly supported, continuous function f, where $${\theta_{y}}$$
 is the shift operator, $${\tau_{f}}$$
 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 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.

Eliran Subag

Papers

The complexity of spherical $p$-spin models—A second moment approach

The geometry of the Gibbs measure of pure spherical spin glasses

Random geometric complexes in the thermodynamic regime

Random geometric complexes in the thermodynamic regime

Freezing and Decorated Poisson Point Processes