Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators

2025/05/22 by Monthus, Cecile · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2505.16356

Abstract

The statistical properties of non-linear observables of the fractal Gaussian field ϕ( x) of negative Hurst exponent H<0 in dimension d are revisited with a focus on spatial-averaging observables and on the properties of the finite parts ϕ_n( x) of the ill-defined composite operators ϕⁿ( x) . For the special case n=2 of quadratic observables, explicit results include the cumulants of arbitrary order, the Lévy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order n>2 is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts ϕ_n( x) of the ill-defined composite operators ϕⁿ( x) and to compute their correlations involving the Hurst exponents H_n=nH.

Citations

• Critical Probability Distributions of the order parameter at two loops I: Ising universality class
• Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations
• A unifying representation of path integrals for fractional Brownian motions
• Rosenblatt distribution subordinated to gaussian random fields with long-range dependence
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Cited by

• Fractional Brownian Motion with Negative Hurst Exponent

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