2009/07/31 by Yan V. Fyodorov, Yan V Fyodorov, Pierre Le Doussal +1 · 2 citations
Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Gaussian #Gaussian free field #Logarithm #Mathematical analysis #Mathematics #Partition function (quantum field theory) #Physics #Quantum mechanics #Random matrix #Statistical Mechanics and Entropy #Statistical mechanics #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Universality (dynamical systems) #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1088/1742-5468/2009/10/p10005
published as J. Stat. Mech. (2009) P10005 · 25 pages, 12 figures Published version. Misprint corrected, references and note added
openalex publication_date 2009/10/01 · arxiv created 2009/10/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
We compute the distribution of the partition functions for a class of one-dimensional random energy models with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/ f noise generated by sampling the two-dimensional Gaussian free field (2D GFF) along various planar curves. Our method extends the recent analysis of Fyodorov and Bouchaud (2008 J. Phys. A: Math. Theor. 41 372001) from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a duality relation satisfied by the suitable generating function of free energy cumulants in the high temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2D GFF on the [0,1] interval. We provide numerical checks of the circular case and the interval case and discuss universality and various extensions. The relevance to the distribution of the length of a segment in Liouville quantum gravity is noted.