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

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**Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs.** / Reddy, Tulasi Ram; Vadlamani, Sreekar; Yogeshwaran, D.

Research output: Contribution to journal › Article

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*Journal of Statistical Physics*, vol. 173, no. 3-4, pp. 941-984. https://doi.org/10.1007/s10955-018-2026-9

### APA

*Journal of Statistical Physics*,

*173*(3-4), 941-984. https://doi.org/10.1007/s10955-018-2026-9

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*Journal of Statistical Physics*. 2018, 173(3-4). 941-984. https://doi.org/10.1007/s10955-018-2026-9

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TY - JOUR

T1 - Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

AU - Reddy, Tulasi Ram

AU - Vadlamani, Sreekar

AU - Yogeshwaran, D.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Cayley graphs

KW - Central limit theorem

KW - Clustering spin models

KW - Cubical complexes

KW - Exponentially quasi-local statistics

KW - Fast decaying covariance

UR - http://www.scopus.com/inward/record.url?scp=85044924708&partnerID=8YFLogxK

U2 - 10.1007/s10955-018-2026-9

DO - 10.1007/s10955-018-2026-9

M3 - Article

AN - SCOPUS:85044924708

VL - 173

SP - 941

EP - 984

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 1572-9613

IS - 3-4

ER -