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 language | English (US) |
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Article number | 76 |
Journal | Electronic Communications in Probability |
Volume | 23 |
DOIs | |
State | Published - 2018 |
Keywords
- Concentration of measure
- Contraction
- Independence testing
- Ising model
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty