Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f1/f Noises generated by Gaussian Free Fields

We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) 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 (2dGFF) along various planar curves. Our method extends the recent analysis of Fyodorov Bouchaud from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a {\it 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 2dGFF on the $[0,1]$ interval. We provide numerical checks of the circular and the interval case and discuss universality and various extensions. Relevance to the distribution of length of a segment in Liouville quantum gravity is noted.Comment: 25 pages, 12 figures Published version. Misprint corrected, references and note adde

Another derivation of the geometrical KPZ relations

We give a physicist's derivation of the geometrical (in the spirit of Duplantier-Sheffield) KPZ relations, via heat kernel methods. It gives a covariant way to define neighborhoods of fractals in 2d quantum gravity, and shows that these relations are in the realm of conformal field theory

Critical curves in conformally invariant statistical systems

We consider critical curves -- conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.Comment: Published versio

Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential

Boltzmann-Gibbs measures generated by logarithmically correlated random potentials are multifractal. We investigate the abrupt change ("pre-freezing") of multifractality exponents extracted from the averaged moments of the measure - the so-called inverse participation ratios. The pre-freezing can be identified with termination of the disorder-averaged multifractality spectrum. Naive replica limit employed to study a one-dimensional variant of the model is shown to break down at the pre-freezing point. Further insights are possible when employing zero-dimensional and infinite-dimensional versions of the problem. In particular, the latter version allows one to identify the pattern of the replica symmetry breaking responsible for the pre-freezing phenomenon.Comment: This is published version, 11 pages, 1 figur

Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model

We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain non-local observables for the two-dimensional Ising model at criticality by combining Smirnov's recent proof of the fact that the scaling limit of critical Ising interfaces can be described by chordal SLE(3) with Kozdron and Lawler's configurational measure on mutually avoiding chordal SLE paths. As an extension of this result, we also compute the probability that an SLE(k) path, k in (0,4], and a Brownian motion excursion do not intersect.Comment: v1: 17 pages, 4 figures, to appear in J. Phys. A: Math. Theor

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small correction

Two Perspectives of the 2D Unit Area Quantum Sphere and Their Equivalence

2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum gravity and is the theory of world-sheet in string theory. Recently there has been growing interest in studying LQG in the realm of probability theory: David, Kupiainen, Rhodes, Vargas (2014) and Duplantier, Miller, Sheffield (2014) both provide a probabilistic perspective of the LQG on the 2D sphere. In particular, in each of them one may find a definition of the so-called unit area quantum sphere. We examine these two perspectives and prove their equivalence by showing that the respective unit area quantum spheres are the same. This is done by considering a unified limiting procedure for defining both objects.Comment: minor revisions; final versio