Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Abstract

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Published in at http://dx.doi.org/10.1214/13-AOP890 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (41)

• Level-Spacing Distributions and the Airy Kernel
• 2D Gravity and Random Matrices
• Liouville Field Theory -- A decade after the revolution
• Random matrices: Universality of local eigenvalue statistics up to the edge
• Convergence in law of the minimum of a branching random walk
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• The largest eigenvalue of rank one deformation of large Wigner matrices
• The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
• Log-infinitely divisible multifractal processes
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Limits of spiked random matrices I
• Multicritical Phases of the O(n) Model on a Random Lattice
• World-Sheet Geometry and Baby Universes in 2-D Quantum Gravity
• Effective dynamics using conditional expectations
• Kramers' law: Validity, derivations and generalisations
• Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields
• Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
• KPZ in one dimensional random geometry of multiplicative cascades
• Strings with Discrete Target Space
• Non-Perturbative Solution of Matrix Models Modified by Trace-Squared Terms
• A Proposal for Strings at D>1
• Densities for Ornstein-Uhlenbeck processes with jumps
• Duality and KPZ in Liouville Quantum Gravity
• Touching Random Surfaces and Liouville Gravity
• Counting colored planar maps: algebraicity results
• Gaussian multiplicative chaos and KPZ duality
• The Potts-q random matrix model : loop equations, critical exponents, and rational case
• Conformal Fractal Geometry and Boundary Quantum Gravity
• Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field
• Lognormal scale invariant random measures
• Multi-spin systems on a randomly triangulated surface
• Q-states Potts model on a random planar lattice
• Critical Mandelbrot Cascades
• A Solvable 2D Quantum Gravity Model with $\GAMMA >0$
• A modified c=1 matrix model with new critical behavior
• Schramm Loewner Evolution and Liouville Quantum Gravity
• Critical behavior in $c=1$ matrix model with branching interactions
• Levy multiplicative chaos and star scale invariant random measures
• Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices
• Boundary Loop Models and 2D Quantum Gravity
• Lipschitz partition processes

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