Concentration inequalities for polynomials of contracting ising models

Reza Gheissari, Eyal Lubetzky, Yuval Peres

Research output: Contribution to journalArticle

Abstract

We study the concentration of a degree-d polynomial of the N spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For d = 1, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related settings by e.g., Chazottes et al. (2007) and Kontorovich and Ramanan (2008). For d = 2, exponential concentration was shown by Marton (2003) on lattices. We treat a general fixed degree d with O(1) coefficients, and show that the polynomial has variance O(Nd) and, after rescaling it by N−d/2, its tail probabilities decay as exp(−c r2/d) for deviations of r ≥ C log N.

Original languageEnglish (US)
Article number76
JournalElectronic Communications in Probability
Volume23
DOIs
StatePublished - Jan 1 2018

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Concentration Inequalities
Ising Model
Polynomial
Glauber Dynamics
Tail Probability
Lipschitz Function
Rescaling
Convex function
Deviation
Decay
Polynomials
Contracting
Coefficient

Keywords

  • Concentration of measure
  • Contraction
  • Independence testing
  • Ising model

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Concentration inequalities for polynomials of contracting ising models. / Gheissari, Reza; Lubetzky, Eyal; Peres, Yuval.

In: Electronic Communications in Probability, Vol. 23, 76, 01.01.2018.

Research output: Contribution to journalArticle

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