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2017

Concentration inequalities for polynomials of contracting Ising models

Gheissari, Reza, Lubetzky, Eyal, Peres, Yuval

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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.

Concentration inequalities for polynomials of contracting Ising models · Around

Built on

A measure concentration inequality for contracting Markov chains
K. Marton · 1996
Earlier work this paper cites.
Trilogy of couplings and general formulas for lower bound of spectral gap
M.-F. Chen · 1998
Earlier work this paper cites.
Concentration of measure inequalities for Markov chains and Φ \Phi -mixing processes
P.-M. Samson · 2000
Earlier work this paper cites.
The concentration of measure phenomenon
M. Ledoux · 2001
Earlier work this paper cites.

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Testing Ising models
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C. Daskalakis, N. Dikkala, and G. Kamath
Cited in the paper.

Then

Concentration inequalities for dependent random variables via the martingale method
L. A. Kontorovich and K. Ramanan · 2008
Later among the works it cites.
Concentration of measure and mixing for Markov chains
M. J. Luczak · 2008
Later among the works it cites.
Markov chains and mixing times
D. A. Levin, Y. Peres, and E. L. Wilmer · 2009
Later among the works it cites.
Gibbs measures and phase transitions
H.-O. Georgii · 2011
Later among the works it cites.

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