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On Barnes Beta Distributions and Applications to the Maximum Distribution of the 2D Gaussian Free Field

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Abstract

A new family of Barnes beta distributions on \((0, \infty )\) is introduced and its infinite divisibility, moment determinacy, scaling, and factorization properties are established. The Morris integral probability distribution is constructed from Barnes beta distributions of types (1, 0) and (2, 2),  and its moment determinacy and involution invariance properties are established. For application, the maximum distributions of the 2D gaussian free field on the unit interval and circle with a non-random logarithmic potential are conjecturally related to the critical Selberg and Morris integral probability distributions, respectively, and expressed in terms of sums of Barnes beta distributions of types (1, 0) and (2, 2).

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Notes

  1. We will abbreviate \(\bigl (\mathcal {S}_N \log \Gamma _M\bigr )(q\,|a,\,b)\) to mean the action of \(\mathcal {S}_N\) on \(\log \Gamma _M(x|a),\) i.e. \(\bigl (\mathcal {S}_N \log \Gamma _M(x|a)\bigr )(q\,|\,b).\)

  2. Theorem 6 in [2] shows that the laws of the total mass of the continuous and discrete multiplicative chaos measures are the same provided the \(\varepsilon \) parameter in (76) coincides with the discretization step.

  3. The validity of approximating the finite N quantity with the \(N\rightarrow \infty \) limit is discussed in [14].

  4. As remarked in [21], this procedure only determines the distribution of the maximum up to a constant term.

  5. This restriction is necessary as \(M_{(\tau ,\lambda _1,\lambda _2)}\) is real-valued whereas \(\int _{-\pi }^{\pi } e^{i\psi \frac{\lambda _1-\lambda _2}{2}} \, |1+e^{i\psi }|^{\lambda _1+\lambda _2}\, dM_{\beta }(\psi )\) is not in general, unless \(\lambda _1=\lambda _2.\) The problem of determining the law of \(\int _{-\pi }^{\pi } e^{i\psi \frac{\lambda _1-\lambda _2}{2}} \, |1+e^{i\psi }|^{\lambda _1+\lambda _2}\, dM_{\beta }(\psi )\) for \(\lambda _1\ne \lambda _2\) is left to future research.

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Acknowledgments

The author wants to express gratitude to Yan Fyodorov and Nickolas Simm for helpful correspondence relating to ref. [21]. The author is also thankful to the referees for many helpful suggestions.

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    Dmitry Ostrovsky

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Ostrovsky, D. On Barnes Beta Distributions and Applications to the Maximum Distribution of the 2D Gaussian Free Field. J Stat Phys 164, 1292–1317 (2016). https://doi.org/10.1007/s10955-016-1591-z

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