Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields

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.

[1]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[2]  E. Saksman,et al.  Random conformal weldings , 2009, 0909.1003.

[3]  Scott Sheffield,et al.  Duality and the Knizhnik-Polyakov-Zamolodchikov relation in Liouville quantum gravity. , 2009, Physical review letters.

[4]  Y. Fyodorov.  Pre-freezing of multifractal exponents in random energy models with a logarithmically correlated potential , 2009, 0903.2502.

[5]  D. Ostrovsky.  Mellin Transform of the Limit Lognormal Distribution , 2009 .

[6]  Scott Sheffield,et al.  Liouville quantum gravity and KPZ , 2008, 0808.1560.

[7]  V. Vargas,et al.  KPZ formula for log-infinitely divisible multifractal random measures , 2008, 0807.1036.

[8]  Peter Morters,et al.  Minimal supporting subtrees for the free energy of polymers on disordered trees , 2008, 0806.3430.

[9]  J. Bouchaud,et al.  Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential , 2008, 0805.0407.

[10]  D. Ostrovsky.  Intermittency Expansions for Limit Lognormal Multifractals , 2008 .

[11]  Z. Rácz,et al.  Finite-size scaling in extreme statistics. , 2007, Physical review letters.

[12]  J. Bouchaud,et al.  Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces , 2007, 0711.4006.

[13]  S. Ole Warnaar,et al.  The importance of the Selberg integral , 2007, 0710.3981.

[14]  Z. Rácz,et al.  Maximal height statistics for 1/f(alpha) signals. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Y. Fyodorov,et al.  Classical particle in a box with random potential: Exploiting rotational symmetry of replicated Hamiltonian , 2006, cond-mat/0610035.

[16]  Olivier Daviaud.  Extremes of the discrete two-dimensional Gaussian free field , 2004, math/0406609.

[17]  Dirk G. A. L. Aarts,et al.  Direct Visual Observation of Thermal Capillary Waves , 2004, Science.

[18]  E. Faleiro,et al.  Theoretical derivation of 1/f noise in quantum chaos. , 2004, Physical review letters.

[19]  F. Schmitt.  A causal multifractal stochastic equation and its statistical properties , 2003, cond-mat/0305655.

[20]  E. Bolthausen,et al.  Entropic repulsion and the maximum of the two-dimensional harmonic crystal , 2001 .

[21]  Victor Adamchik,et al.  On the Barnes function , 2001, ISSAC '01.

[22]  D. Carpentier,et al.  Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  E. Bacry,et al.  Multifractal random walk. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  V. Fateev,et al.  Boundary Liouville Field Theory I. Boundary State and Boundary Two-point Function , 2000, hep-th/0001012.

[25]  P. Francesco,et al.  Conformal Field Theory , 1999 .

[26]  P. Goldbart,et al.  Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field , 1997, cond-mat/9706084.

[27]  Mudry,et al.  Localization in Two Dimensions, Gaussian Field Theories, and Multifractality. , 1996, Physical review letters.

[28]  O. Zeitouni,et al.  Entropic repulsion of the lattice free field , 1995 .

[29]  B. Derrida,et al.  Polymers on disordered trees, spin glasses, and traveling waves , 1988 .

[30]  P. G. de Gennes,et al.  A model for contact angle hysteresis , 1984 .

[31]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[32]  Albrecht Böttcher,et al.  Spectral properties of banded Toeplitz matrices , 1987 .