On the Universality of the Superconcentration in Mixed p-Spin Models

Can, Van Hao; Nguyen, Van Quyet; Vu, Hong-Son

doi:10.1007/s10955-023-03093-8

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Introduction and Main Results1

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2023

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“…Remark Under the assumption that the first four moments of

h

agree with those of

g

and

h

has a finite fifth moment, one can manage to match the first and second moments of the free energies associated to

h

and

g

asymptotically by using the approximate Gaussian integration by parts, which will lead to

\mathrm{Var}\left({F}_N\right)=O\left(N/\log N\right)

as () and (), see, for example, [9]. Our main results address superconcentration by reducing the moment assumption as much as possible.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universality of superconcentration in the Sherrington–Kirkpatrick model

Chen

Lam

2023

Random Struct Algorithms

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We study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order , in contrast to the bound obtained from the Gaussian–Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function that arises in the representation of the disorder as for standard normal. Under an additional regularity assumption on , we further show that the variance is of order at most .

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“…Remark Under the assumption that the first four moments of