scholarly journals A Central Limit Theorem for Weighted Averages of Spins in the High Temperature Region of the Sherrington-Kirkpatrick Model

2005 ◽  
Vol 10 (0) ◽  
pp. 499-524
Author(s):  
Dmitry Panchenko
Keyword(s):  
High Temperature ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Temperature Region ◽  
High Temperature Region ◽  
Weighted Averages ◽  
Kirkpatrick Model

scholarly journals Local Central Limit Theorem for Multi-group Curie–Weiss Models

2021 ◽  
Author(s):  
Michael Fleermann ◽  
Werner Kirsch ◽  
Gabor Toth
Keyword(s):  
High Temperature ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Temperature Regime ◽  
Mean Field ◽  
High Temperature Regime ◽  
The Mean ◽  
Local Central Limit Theorem ◽  
Group Version

AbstractWe study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is, for the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high-temperature regime.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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