On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces

Abstract

We construct a N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension N>>1 the free energy of the system in the thermodynamic limit coincides with the most general version of Derrida's Generalized Random Energy Model. The low-temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape. We argue that our construction is in fact valid in any finite spatial dimensions, starting from N=1.

arXiv comment4 pages, no figures

References in corpus (7)

• Models of traps and glass phenomenology
• A multifractal random walk
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Separation of time and length scales in spin-glasses: temperature as a microscope
• Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field
• Large times off-equilibrium dynamics of a particle in a random potential
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian

Cited by in corpus (7)

• Extreme value statistics of correlated random variables: a pedagogical review
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality
• Liouville field theory and log-correlated Random Energy Models
• One step replica symmetry breaking and extreme order statistics of logarithmic REMs
• Random cascade models of multifractality : real-space renormalization and travelling-waves