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There are many classical random walk in random environment results that apply to ergodic random planar environments.
Remarques sur la convergence en loi des martingales vers des martingales continues
R. Rebolledo · 1977
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Quantum geometry of bosonic strings
A. M. Polyakov · 1981
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Quantum geometry of fermionic strings
A. M. Polyakov · 1981
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Central limit theorems for martingales with discrete or continuous time
I. S. Helland · 1982
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Sur le chaos multiplicatif
J.-P. Kahane · 1985
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Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions
C. Kipnis and S. R. S. Varadhan · 1986
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An invariance principle for reversible Markov processes. Applications to random motions in random environments
A. De Masi, P. A. Ferrari, S. Goldstein, and W. D. Wick · 1989
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Generating random spanning trees more quickly than the cover time
D. B. Wilson · 1996
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Diffusions on fractals
M. T. Barlow · 1998
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Hölder regularity and dimension bounds for random curves
M. Aizenman and A. Burchard · 1999
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Percolation
G. Grimmett · 1999
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Volume of metric balls in Liouville quantum gravity
M. Ang, H. Falconet, and X. Sun · 2001
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Quenched invariance principles for walks on clusters of percolation or among random conductances
V. Sidoravicius and A.-S. Sznitman · 2004
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Processes on unimodular random networks
D. Aldous and R. Lyons · 2007
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Quenched invariance principle for simple random walk on percolation clusters
N. Berger and M. Biskup · 2007
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Functional CLT for random walk among bounded random conductances
M. Biskup and T. M. Prescott · 2007
Cited alongside, same era.
Invariance principle for the random conductance model with unbounded conductances
M. T. Barlow and J.-D. Deuschel · 2010
Cited alongside, same era.
Probability: theory and examples
R. Durrett · 2010
Cited alongside, same era.
Random walks on disordered media and their scaling limits
T. Kumagai · 2010
Cited alongside, same era.
Recent progress on the random conductance model
M. Biskup · 2011
Cited alongside, same era.
Liouville quantum gravity and KPZ
B. Duplantier and S. Sheffield · 2011
C. Garban, R. Rhodes, and V. Vargas · 2016
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Probability on Trees and Networks
R. Lyons and Y. Peres · 2016
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Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding
J. Miller and S. Sheffield · 2016
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Liouville quantum gravity and the Brownian map III: the conformal structure is determined
J. Miller and S. Sheffield · 2016
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Quenched invariance principle for simple random walk on clusters in correlated percolation models
E. B. Procaccia, R. Rosenthal, and A. Sapozhnikov · 2016
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Fluctuations in Markov processes
T. Komorowski, C. Landim, and S. Olla · 2012
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Invariance principle for the random conductance model
S. Andres, M. T. Barlow, J.-D. Deuschel, and B. M. Hambly · 2013
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Recurrence of planar graph limits
O. Gurel-Gurevich and A. Nachmias · 2013
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Liouville quantum gravity as a mating of trees
B. Duplantier, J. Miller, and S. Sheffield · 2014
Cited alongside, same era.
Gaussian multiplicative chaos and applications: A review
R. Rhodes and V. Vargas · 2014
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Diffusion in planar Liouville quantum gravity
N. Berestycki · 2015
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Conformal weldings of random surfaces: SLE and the quantum gravity zipper
S. Sheffield · 2016
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Conformal growth rates and spectral geometry on distributional limits of graphs
J. R. Lee · 2017
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Liouville quantum gravity on the unit disk
Y. Huang, R. Rhodes, and V. Vargas · 2018
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Quasisymmetric uniformization and heat kernel estimates
M. Murugan · 2018
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Harmonic functions on mated-CRT maps
E. Gwynne, J. Miller, and S. Sheffield · 2019
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Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field
M. Biskup, J. Ding, and S. Goswami · 2020
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E. Gwynne, J. Miller, and S. Sheffield · 2020
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Liouville quantum gravity and the Brownian map I: the QLE ( 8 / 3 , 0 ) {\rm QLE}(8/3,0) metric
J. Miller and S. Sheffield · 2020
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The tutte embedding of the mated-crt map converges to liouville quantum gravity
E. Gwynne, J. Miller, and S. Sheffield · 2021
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