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

Reddy, Tulasi Ram; Vadlamani, Sreekar LU and Yogeshwaran, D. (2018) In Journal of Statistical Physics 173(3-4). p.941-984
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... (More)

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 (Formula presented.)-mixing (for local statistics) and exponential (Formula presented.)-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.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Cayley graphs, Central limit theorem, Clustering spin models, Cubical complexes, Exponentially quasi-local statistics, Fast decaying covariance
in
Journal of Statistical Physics
volume
173
issue
3-4
pages
941 - 984
publisher
Springer
external identifiers
  • scopus:85044924708
ISSN
0022-4715
DOI
10.1007/s10955-018-2026-9
language
English
LU publication?
yes
id
3b0fe222-1d89-45c7-be47-4465f058610d
date added to LUP
2018-04-18 16:21:37
date last changed
2019-01-14 17:37:33
@article{3b0fe222-1d89-45c7-be47-4465f058610d,
  abstract     = {<p>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 (Formula presented.)-mixing (for local statistics) and exponential (Formula presented.)-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.</p>},
  author       = {Reddy, Tulasi Ram and Vadlamani, Sreekar and Yogeshwaran, D.},
  issn         = {0022-4715},
  keyword      = {Cayley graphs,Central limit theorem,Clustering spin models,Cubical complexes,Exponentially quasi-local statistics,Fast decaying covariance},
  language     = {eng},
  month        = {04},
  number       = {3-4},
  pages        = {941--984},
  publisher    = {Springer},
  series       = {Journal of Statistical Physics},
  title        = {Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs},
  url          = {http://dx.doi.org/10.1007/s10955-018-2026-9},
  volume       = {173},
  year         = {2018},
}