Minimum of Random Energy Model (REM) with logarithmically correlated potential

Question

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of the minimum in a suitable asymptotic regime is given by $$ 2\mathrm{e}^{\beta_c x/2}\operatorname{K}_1\left(2\mathrm{e}^{\beta_c x/2}\right)\tag{*}, $$ where $\operatorname{K}_1$ is the modified Bessel function of the second kind (MacDonald function) of order 1, and $\beta_c=1/g$ is the inverse of the critical temperature of the model.

The model itself is defined in Section 3.1. as a collection of mean-zero Gaussian random variables $(V_i)_{i=1,\ldots,M}$ with covariance $$ \langle V_i,V_j\rangle = -g^2\log\left(4\sin^2\frac{\theta_i-\theta_j}{2}\right),\quad \theta_i= \frac{2\pi\,i}{M}. $$

In the conclusion the authors write that "Although [they] think [their] results are supported by rather convincing arguments, the calculations are very essentially based on a few plausible but not yet fully verified assumptions."

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Question: Has this conjectured form (*) of the limiting distribution been established rigorously?

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[FB] Fyodorov, Yan V.; Bouchaud, Jean-Philippe. Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 (2008), no. 37, 372001, 12 pp. MR2430565 (2010g:82036)

Answer 1

The Bessel-function distribution you quote is the convolution of two Gumbel distributions. This form for the extreme value statistics was rigorously derived for a broad class of stochastic processes by Subag and Zeitouni, Freezing and decorated Poisson point processes (2014). The log-correlated process of Fyodorov and Bouchaud is not in that class, so the best we have at the moment is numerical evidence that the Gumbel distribution still applies.