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Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

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Abstract

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the Lp functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the Lp norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p 0 = 2+4π2/β2ω 2 , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p 0 and p > p 0 . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.

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  1. Lomonosov Moscow State University, Moscow, Russia

    V. R. Fatalov

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 168, No. 2, pp. 299–340, August, 2011.

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Fatalov, V.R. Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure. Theor Math Phys 168, 1112–1149 (2011). https://doi.org/10.1007/s11232-011-0092-0

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