Erratum : Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models [Phys. Rev. E 63, 026110 (2001)]

Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

“…Both the decrease rate and the short correlation range ensure that the distribution of converge to a Gumbel distribution [50][51] However the convergence may be slow as discussed above ( figure 12a). This is the reason why we restricted the range of β in figure 11 to (0-4).…”

Joint min-max distribution and Edwards-Anderson's order parameter of the circular 1/f-noise model

“…Our results provide correction to this approximation for the circular model. Another highlight is the modification of the amplitude of the joint Carpentier-Le Doussal (CLD) tail [3] by the max/min correlation (compared to product of marginals). We shall give a formula (13) for the tail ratio for general log-REM's.…”

“…The statistical physics of a particle in logarithmically correlated random potentials was initially studied as simplified models of spin glass known as logarithmic Random Energy Models (log-REM's) [1][2][3], but is now realised to be relevant to subjects ranging from multi-fractal wavefunctions [4,5], extrema of 2d Gaussian Free Field (GFF) [6,7] and 2d quantum gravity [8], to the value distribution of random matrix characteristic polynomials and the Riemann zeta on the critical line [9][10][11].…”

“…The minimum and maximum, denoted as V M± = ± min M j=1 (±V j,M ), are known for standard log-REM's to satisfy [3,29] …”

“…The extension of freezing to describe the free energy fluctuation has a long history [2,3,12,13] and was recently promoted to a rigorous stage [14] by using derivative multiplicative chaos [8,15]. Yet, explicit predictions (e.g., of free energy distribution) still require non-rigorous approaches and some "integrability" coming from log-gas integrals [16], β-random matrix theory [17], or symmetric functions [18].…”