TY - JOUR

T1 - Liouville dynamical percolation

AU - Garban, Christophe

AU - Holden, Nina

AU - Sepúlveda, Avelio

AU - Sun, Xin

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/8

Y1 - 2021/8

N2 - We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a γ-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the γ-Liouville measure associated with a two-dimensional log-correlated field h. Roughly speaking, this continuum percolation process evolves very rapidly where the field h is high and barely moves where the field h is low. Our main results can be summarized as follows.First, we build this inhomogeneous dynamical percolation, which we call γ-Liouville dynamical percolation (LDP), by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: γ∈[0,2-5/2), γ∈[2-5/2,3/2), and γ∈(3/2,2).When γ<3/2, we prove that γ-LDP is mixing in the Schramm–Smirnov space as t→ ∞, quenched in the log-correlated field h. On the contrary, when γ>3/2 the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for γ=1/6 is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations.When γ<3/4, we obtain quantitative bounds on the mixing of quad crossing events.

AB - We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a γ-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the γ-Liouville measure associated with a two-dimensional log-correlated field h. Roughly speaking, this continuum percolation process evolves very rapidly where the field h is high and barely moves where the field h is low. Our main results can be summarized as follows.First, we build this inhomogeneous dynamical percolation, which we call γ-Liouville dynamical percolation (LDP), by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: γ∈[0,2-5/2), γ∈[2-5/2,3/2), and γ∈(3/2,2).When γ<3/2, we prove that γ-LDP is mixing in the Schramm–Smirnov space as t→ ∞, quenched in the log-correlated field h. On the contrary, when γ>3/2 the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for γ=1/6 is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations.When γ<3/4, we obtain quantitative bounds on the mixing of quad crossing events.

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U2 - 10.1007/s00440-021-01057-1

DO - 10.1007/s00440-021-01057-1

M3 - Article

AN - SCOPUS:85107295441

SN - 0178-8051

VL - 180

SP - 621

EP - 678

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -