scholarly journals Local Central Limit Theorem for Determinantal Point Processes

2014 ◽  
Vol 157 (1) ◽  
pp. 60-69 ◽  
Author(s):  
Peter J. Forrester ◽  
Joel L. Lebowitz
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Point Processes ◽  
Determinantal Point Processes ◽  
Local Central Limit Theorem

scholarly journals Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

2021 ◽  
Vol 179 (3-4) ◽  
pp. 1145-1181 ◽  
Author(s):  
Sebastian Andres ◽  
Alberto Chiarini ◽  
Martin Slowik
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Local Limit Theorem ◽  
Time Dependent ◽  
Moment Condition ◽  
Difference Operators ◽  
Random Conductance Model ◽  
Local Central Limit Theorem ◽  
Random Conductance

AbstractWe establish a quenched local central limit theorem for the dynamic random conductance model on $${\mathbb {Z}}^d$$ Z d only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi’s iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights.

scholarly journals Gaussian fluctuations of Young diagrams under the Plancherel measure

2007 ◽  
Vol 463 (2080) ◽  
pp. 1069-1080 ◽  
Author(s):  
Leonid V Bogachev ◽  
Zhonggen Su
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Gaussian Process ◽  
Central Limit ◽  
Point Processes ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Finite Dimensional ◽  
The Central Limit Theorem ◽  
Local Correlations

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions λ ⊢ n ∈ (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like , the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin–Lebowitz–Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45°, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.

scholarly journals Quenched local central limit theorem for random walks in a time-dependent balanced random environment

2021 ◽  
Author(s):  
Jean-Dominique Deuschel ◽  
Xiaoqin Guo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Transition Probabilities ◽  
Time Dependent ◽  
Upper And Lower Bounds ◽  
Negative Moment ◽  
Local Central Limit Theorem

AbstractWe prove a quenched local central limit theorem for continuous-time random walks in $${\mathbb {Z}}^d, d\ge 2$$ Z d , d ≥ 2 , in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.

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