Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

Abstract

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of α-mixing (for local statistics) and exponential α-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.Keywords Clustering spin models · mixing random fields · central limit theorem · Cayley graphs · fast decaying covariance · Gaussian random field · determinantal point process · exponentially quasi-local statistics · cubical complexes · nearest-neighbour graphs Mathematics Subject Classification (2000) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs · 60G60 Random fields · 60F05 Central limit theorems and other weak theorems · 60D05 Geometric probability and stochastic geometry.

Normal convergence of nonlocalised geometric functionals and shot-noise excursions

“…where F 0 is some functional on the class of subsets of Z d , with finite second moment under iid Bernoulli input. Stabilising functionals and excursions functionals yield possible applications, our findings might apply for instance to the results of [22], where more general classes of discrete input than Bernoulli processes are also treated. Seeing F W (or F ′ W ) as a functional of η, the variance and asymptotic normality results of Theorems 1.1-1.2 apply to F W under conditions of the type…”

Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures

“…However, if T 1 = T 2 such a result is not known, it can only be achieved for domains such that asymptotically dist( T 1 , T 2 ) → ∞ (see the proof of Proposition 3). Indeed, we are not aware of any joint central limit theorem for the LK densities of excursion sets of stationary random fields on , except in the case of a Boolean model in Hug et al (2016) or in the case of pixelated images in Reddy et al (2018) and binary images in Ebner et al (2018). Such a constraint on T 1 and T 2 is satisfied for instance taking T 1 ∪ T 2 ⊂ T such that T 1 ∩ T 2 = ∅ , | T 1 |= | T 2 |, | T 1 | > c | T |, 0 < c < 1/2 and having From a practical point of view, one only has one (large) image, the inference procedure can be implemented by partitioning it in subwindows T 1 and T 2 separated by a “large” band (for instance of width ) in order to build the estimator .…”

Betti Numbers of Gaussian Excursions in the Sparse Regime

“…In random graphs terminology, this places all our key results in the 'sparse' regime. The asymptotics in [14,27,31,35], from the perspective of our setup, loosely translates to keeping u fixed and letting only n increase to infinity. It is easy to see that the average vertex degree would asymptotically then be a constant.…”

“…LKCs do provide useful topological information, but not at the level of individual Betti numbers. In relation to Betti numbers, the only paper of which we are aware is [35]. There, a weak law and a (multivariate) CLT have been shown for a generic class of (quasi-) local statistics of spin models on Cayley graphs.…”