1,028 research outputs found
Estimation in spin glasses: A first step
The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of
neural networks and the Ising spin glass are all models of binary data
belonging to the one-parameter exponential family with quadratic sufficient
statistic. Under bare minimal conditions, we establish the
-consistency of the maximum pseudolikelihood estimate of the natural
parameter in this family, even at critical temperatures. Since very little is
known about the low and critical temperature regimes of these extremely
difficult models, the proof requires several new ideas. The author's version of
Stein's method is a particularly useful tool. We aim to introduce these
techniques into the realm of mathematical statistics through an example and
present some open questions.Comment: Published in at http://dx.doi.org/10.1214/009053607000000109 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Absence of replica symmetry breaking in the random field Ising model
It is shown that replica symmetry is not broken in the random field Ising
model in any dimension, at any temperature and field strength, except possibly
at a measure-zero set of exceptional temperatures and field strengths.Comment: 11 pages. To appear in Commun. Math. Phy
A generalization of the Lindeberg principle
We generalize Lindeberg's proof of the central limit theorem to an invariance
principle for arbitrary smooth functions of independent and weakly dependent
random variables. The result is applied to get a similar theorem for smooth
functions of exchangeable random variables. This theorem allows us to identify,
for the first time, the limiting spectral distributions of Wigner matrices with
exchangeable entries.Comment: Published at http://dx.doi.org/10.1214/009117906000000575 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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