Critical Liouville measure as a limit of subcritical measures

Abstract

We study how the Gaussian multiplicative chaos (GMC) measures ${\mu }^{\gamma }$ corresponding to the 2D Gaussian free field change when $\gamma$ approaches the critical parameter $2$. In particular, we show that as $\gamma \to 2^{-}$, $(2-\gamma {)}^{-1}{\mu }^{\gamma }$ converges in probability to $2{\mu }^{\prime }$, where ${\mu }^{\prime }$ is the critical GMC measure.

arXiv comment15 pages, no figures needed

References in corpus (7)

• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity
• Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation
• The first passage sets of the 2D Gaussian free field: convergence and isomorphisms
• Gaussian multiplicative chaos through the lens of the 2D Gaussian free field
• First passage sets of the 2D continuum Gaussian free field
• Liouville Quantum Gravity on the unit disk

Cited by in corpus (19)

• The Fyodorov-Bouchaud formula and Liouville conformal field theory
• First passage sets of the 2D continuum Gaussian free field
• On intermediate level sets of two-dimensional discrete Gaussian Free Field
• Conformal welding for critical Liouville quantum gravity
• Upper bounds on Liouville first passage percolation and Watabiki's prediction
• Subsequential scaling limits for Liouville graph distance
• Critical Gaussian multiplicative chaos: a review
• Nesting statistics in the O(n) loop model on random planar maps
• Stochastic quantization of Liouville conformal field theory
• Critical Brownian multiplicative chaos
• The distribution of Gaussian multiplicative chaos on the unit interval
• Conformal welding of quantum disks
• Maxima of log-correlated fields: some recent developments
• Tail universality of critical Gaussian multiplicative chaos
• Dimension Results for the Spectral Measure of the Circular Beta Ensembles
• Liouville measure as a multiplicative cascade via level sets of the Gaussian free field
• Mating of trees for critical Liouville quantum gravity
• An elementary approach to quantum length of SLE
• On the $\log$-regularity of SLE${}_4$