A general method for lower bounds on fluctuations of random variables

Chatterjee, Sourav

doi:10.1214/18-aop1304

Cited by 29 publications

(37 citation statements)

References 65 publications

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“…On the issue of the limiting variance σ M , it is clear that a necessary condition for the variance to be nondegenerate is that P(ω e − ν v − ν u > 0) > 0, otherwise the empty matching is optimal. The author in [5] establishes some general conditions for obtaining fluctuation lower bounds, and one of the techniques can be adapted for the ground state energy under appropriate conditions on the edge and vertex weights. 1.3.…”

Section: Theorem 12 (Central Limit Theorem For the Log-partition Func...mentioning

confidence: 99%

Disordered Monomer-Dimer model on Cylinder graphs

Dey¹,

Krishnan²

2021

Preprint

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We consider the disordered monomer-dimer model on cylinder graphs Gn, i.e., graphs given by the Cartesian product of the line graph on n vertices, and a deterministic graph. The edges carry i.i.d. random weights, and the vertices also carry i.i.d. random weights, not necessarily from the same distribution. Given the random weights, we define a Gibbs measure on the space of monomer-dimer configurations on Gn. We show that the associated free energy converges to a limit, and with suitable scaling and centering, satisfies a central limit theorem. We also show that the number of monomers in a typical configuration satisfies a law of large numbers and a central limit theorem with appropriate centering and scaling. Finally, for an appropriate height function associated with a matching, we show convergence to a limiting function and prove the Brownian motion limit about the limiting height function in the sense of finite-dimensional distributions.

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Section: Theorem 12 (Central Limit Theorem For the Log-partition Func...mentioning

confidence: 99%

Disordered Monomer-Dimer model on Cylinder graphs

Dey¹,

Krishnan²

2021

Preprint

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“…To the best of our limited knowledge, very few things are known about the fluctuations of the free energy in this regime. One might look at Chatterjee [7] where it is proved that the fluctuation of the free energy of the Sherrington-Kirkpatrick model is at least 0(1). When the spins are uniformly distributed on S n−1 , the free energy analogously undergoes a phase transition at β = 1 2 .…”

Section: The Model Descriptionmentioning

confidence: 99%

Fluctuation of the Free Energy of Sherrington–Kirkpatrick Model with Curie–Weiss Interaction: The Paramagnetic Regime

Banerjee

2019

J Stat Phys

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We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by Baik and Lee [3] and Baik et al. [4] which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to Baik and Lee [3] in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by Banerjee [5]. The proof of the approximation by the linear spectral statistics part is taken from Banerjee and Ma [6].

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“…Another route towards anti-concentration is by a coupling approach: when the variable of interest is a function of a random environment, one can often couple two instances of the environment so that one instance of the variable is larger than the other. This approach was taken by Wehr and Aizenman [36] to yield lower bounds on certain variances in the context of the Ising model (and other related models) and is also taken up in other ad-hoc approaches to proving lower bounds on fluctuations [4,14,16,17,18,20,32] which culminated in a recent unifying work of Chatterjee [6].…”

Section: Introductionmentioning

confidence: 99%

Anti-concentration of random variables from zero-free regions

Michelen¹,

Sahasrabudhe²

2021

Preprint

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This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let X be a random variable taking values in {0, . . . , n} with P(X = 0)P(X = n) > 0 and with probability generating function f X . We show that if all of the zeroswhere c > 0 is a absolute constant. We show that this result is sharp, up to the factor 2 in the exponent of R. As a consequence, we are able to deduce a Littlewood-Offord type theorem for random variables that are not necessarily sums of i.i.d. random variables.

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A general method for lower bounds on fluctuations of random variables

Cited by 29 publications

References 65 publications

Disordered Monomer-Dimer model on Cylinder graphs

Disordered Monomer-Dimer model on Cylinder graphs

Fluctuation of the Free Energy of Sherrington–Kirkpatrick Model with Curie–Weiss Interaction: The Paramagnetic Regime

Anti-concentration of random variables from zero-free regions

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