In the paper [FB] ([ArXiv](http://arxiv.org/abs/0805.0407), [J. Phys. A](http://iopscience.iop.org/1751-8121/41/37/372001)), 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](http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html) 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](http://www.ams.org/mathscinet-getitem?mr=2430565) (2010g:82036)



Minimum of Random Energy Model (REM) with logarithmically correlated potential
The Bessel-function distribution you quote is the convolution of two [Gumbel distributions](https://en.wikipedia.org/wiki/Gumbel_distribution). 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](http://arxiv.org/abs/1404.7346) (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](http://arxiv.org/abs/1211.6063) that the Gumbel distribution still applies.


