TY - JOUR

T1 - Liouville dynamical percolation

AU - Garban, Christophe

AU - Holden, Nina

AU - Sepúlveda, Avelio

AU - Sun, Xin

N1 - Funding Information:
We thank Jeffrey Steif for a very good comment and discussion after our talk in Oberwolfach, which lead to Sect. . We also thank Rick Bradley and Jean-Paul Thouvenot for their inputs to that section, we thank Ewain Gwynne for helpful comments to the paper, and we thank the anonymous referee for many helpful comments and careful reading of the paper. The research of C.G. is supported by the ERC grant LiKo 676999 and Institut Universitaire de France (IUF). The research of N.H. is supported by Dr. Max Rössler, the Walter Haefner Foundation, the ETH Zürich Foundation, and a fellowship from the Norwegian Research Council. The research of A.S was supported by the ERC grant LiKo 676999 and is now supported by Grant ANID AFB170001 and FONDECYT iniciación de investigación No 11200085. The research of X.S. was supported by Simons Society of Fellows under Award 527901, and by NSF Award DMS-1811092 and DMS-2027986. Part of the work on this paper was carried out during the visit of N.H. and X.S. to Lyon in November 2017 and 2018. They thank for the hospitality and for the funding through the ERC grant LiKo 676999. A.S would also like to thank the hospitality of Núcleo Milenio “Stochastic models of complex and disordered systems” for repeated invitation to Santiago, where part of this paper was written.
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.

UR - http://www.scopus.com/inward/record.url?scp=85107295441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85107295441&partnerID=8YFLogxK

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 -