Recent progress on the Random Conductance Model

Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.

[1]  H. Spohn.  SCALING LIMIT FOR A CLASS OF GRADIENT FIELDS WITH NON-CONVEX POTENTIALS BY MAREK BISKUP , 2010 .

[2]  S. Varadhan,et al.  Large deviations , 2019, Graduate Studies in Mathematics.

[3]  T. Liggett.  Continuous Time Markov Processes: An Introduction , 2010 .

[4]  J. Wehr.  A lower bound on the variance of conductance in random resistor networks , 1997 .

[5]  R. Durrett.  Probability: Theory and Examples , 1993 .

[6]  Harry Kesten,et al.  First-passage percolation, network flows and electrical resistances , 1984 .

[7]  Y. Peres,et al.  Evolving sets, mixing and heat kernel bounds , 2003, math/0305349.

[8]  D. Stroock,et al.  A new proof of Moser's parabolic harnack inequality using the old ideas of Nash , 1986 .

[9]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[10]  A. V. D. Vaart,et al.  Lectures on probability theory and statistics , 2002 .

[11]  G. Giacomin.  Limit Theorems for Random Interface Models of Ginzburg-Landau ∇ φ type , 2002 .

[12]  F. Otto,et al.  An optimal variance estimate in stochastic homogenization of discrete elliptic equations , 2011, 1104.1291.

[13]  Stefano Olla.  Central limit theorems for tagged particles and for diffusions in random environment , 2001 .

[14]  T. K. Carne,et al.  A transmutation formula for Markov chains , 1985 .

[15]  S. Sheffield.  Gaussian free fields for mathematicians , 2003, math/0312099.

[16]  Alexander Grigor.  HEAT KERNEL UPPER BOUNDS ON A COMPLETE NON-COMPACT MANIFOLD , 1994 .

[17]  S. Varadhan,et al.  Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions , 1986 .

[18]  Harmonic deformation of Delaunay triangulations , 2010, 1012.1677.

[19]  Omar Boukhadra.  Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances Model , 2009, 0907.4525.

[20]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[21]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[22]  Y. Derriennic.  Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem'' , 2006 .

[23]  A. Grigor’yan.  Heat kernels on metric measure spaces with regular volume growth , 2009 .

[24]  Marek Biskup,et al.  Anomalous heat-kernel decay for random walk among bounded random conductances , 2006, ArXiv.

[25]  A. Sokal,et al.  Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality , 1988 .

[26]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[27]  Sunder Sethuraman,et al.  Diffusive limit of a tagged particle in asymmetric simple exclusion processes , 2000 .

[28]  Michael Woodroofe,et al.  Central limit theorems for additive functionals of Markov chains , 2000 .

[29]  S. Olla,et al.  EQUILIBRIUM FLUCTUATIONS FOR ∇ϕ INTERFACE MODEL , 2001 .

[30]  N. Varopoulos,et al.  Long range estimates for markov chains , 1985 .

[31]  H. Kesten,et al.  On birth and death processes in symmetric random environment , 1984 .

[32]  Houman Owhadi,et al.  Approximation of the effective conductivity of ergodic media by periodization , 2002 .

[33]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[34]  Rolf Künnemann,et al.  The diffusion limit for reversible jump processes onZd with ergodic random bond conductivities , 1983 .

[35]  R. Schonmann,et al.  Domination by product measures , 1997 .

[36]  Scott Sheffield,et al.  Random Surfaces , 2003, math/0304049.

[37]  Daniel W. Stroock,et al.  A New Proof of Moser's Parabolic Harnack Inequality via the Old Ideas of Nash , 2022 .

[38]  Magda Peligrad,et al.  Central limit theorem for stationary linear processes , 2005, math/0509682.

[39]  F. Spitzer.  Principles Of Random Walk , 1966 .

[40]  B. M. Brown,et al.  Martingale Central Limit Theorems , 1971 .

[41]  M. Biskup,et al.  Quenched invariance principle for simple random walk on percolation clusters , 2005, math/0503576.

[42]  Christopher Hoffman,et al.  Return Probabilities of a Simple Random Walk on Percolation Clusters , 2005 .

[43]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .

[44]  M. Biskup,et al.  Subdiffusive heat‐kernel decay in four‐dimensional i.i.d. random conductance models , 2010, J. Lond. Math. Soc..

[45]  Olivier Daviaud.  Extremes of the discrete two-dimensional Gaussian free field , 2004, math/0406609.

[46]  M. Eisen,et al.  Probability and its applications , 1975 .

[47]  Asaf Nachmias,et al.  The Alexander-Orbach conjecture holds in high dimensions , 2008, 0806.1442.

[48]  Jeremy Quastel,et al.  Diffusion of color in the simple exclusion process , 1992 .

[49]  Jason Miller.  Universality for SLE(4) , 2010, 1010.1356.

[50]  A. Grigor’yan,et al.  Pointwise Estimates for Transition Probabilities of Random Walks on Infinite Graphs , 2003 .

[51]  A. Yehudayoff,et al.  Loop-erased random walk and Poisson kernel on planar graphs , 2008, 0809.2643.

[52]  Quantitative Version Of The Kipnis-Varadhan Theorem And Monte Carlo Approximation Of Homogenized Coefficients , 2011, 1103.4591.

[53]  É. Remy,et al.  Isoperimetry and heat kernel decay on percolation clusters , 2003, math/0301213.

[54]  H. Kesten.  Percolation theory for mathematicians , 1982 .

[55]  P. Ferrari,et al.  An invariance principle for reversible Markov processes. Applications to random motions in random environments , 1989 .

[56]  P. Caputo,et al.  Invariance principle for Mott variable range hopping and other walks on point processes , 2009, 0912.4591.

[57]  Tadahisa Funaki,et al.  Stochastic Interface Models , 2005 .

[58]  G. Lawler.  Intersections of random walks , 1991 .

[59]  N. Dms.  Functional CLT for random walk among bounded random conductances , 2007 .

[60]  Jérôme Dedecker,et al.  On the functional central limit theorem for stationary processes , 2000 .

[61]  J. Chayes,et al.  Bulk transport properties and exponent inequalities for random resistor and flow networks , 1986 .

[62]  Clement Rau,et al.  Sur le nombre de points visit\'{e}s par une marche al\'{e}atoire sur un amas infini de percolation , 2006, math/0605056.

[63]  J. Mourrat.  A quantitative central limit theorem for the random walk among random conductances , 2011, 1105.4485.

[64]  S. Alexander,et al.  Density of states on fractals : « fractons » , 1982 .

[65]  Russell Lyons,et al.  Biased random walks on Galton–Watson trees , 1996 .

[66]  L. Dubins.  On a Theorem of Skorohod , 1968 .

[67]  G. Grimmett,et al.  The supercritical phase of percolation is well behaved , 1990, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[68]  On the exactness of the Wu-Woodroofe approximation , 2009 .

[69]  Andrey L. Piatnitski,et al.  Quenched invariance principles for random walks on percolation clusters , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[70]  M. Barlow.  Random walks on supercritical percolation clusters , 2003, math/0302004.

[71]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[72]  Ofer Zeitouni,et al.  A central limit theorem for biased random walks on Galton–Watson trees , 2006 .

[73]  E. Davies.  Large Deviations for Heat Kernels on Graphs , 1993 .

[74]  N. Varopoulos.  Isoperimetric inequalities and Markov chains , 1985 .

[75]  Y. Velenik.  Localization and delocalization of random interfaces , 2005, math/0509695.

[76]  T. Schrøder,et al.  ac Hopping conduction at extreme disorder takes place on the percolating cluster. , 2008, Physical review letters.

[77]  S. Popov,et al.  Random walks on Galton–Watson trees with random conductances , 2011, 1101.2769.

[78]  Noam Berger,et al.  A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment , 2011 .

[79]  Gregory F. Lawler,et al.  Weak convergence of a random walk in a random environment , 1982 .

[80]  P. Mathieu,et al.  On symmetric random walks with random conductances on ℤd , 2004 .

[81]  G. Faraud.  A Central Limit Theorem for Random Walk in a Random Environment on a Marked Galton-Watson Tree. , 2011 .

[82]  A. Telcs,et al.  Diffusive Limits on the Penrose Tiling , 2009, 0910.4296.

[83]  Ravi Montenegro,et al.  Mathematical Aspects of Mixing Times in Markov Chains , 2006, Found. Trends Theor. Comput. Sci..

[84]  D. Volný.  Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem , 2009, 0912.2864.

[85]  C. C. Heyde.  Central Limit Theorem , 2006 .

[86]  V. Sidoravicius,et al.  Quenched invariance principles for walks on clusters of percolation or among random conductances , 2004 .

[87]  Y. Peres,et al.  Thick points of the Gaussian free field. , 2009, 0902.3842.

[88]  Phase coexistence of gradient Gibbs states , 2005, math/0512502.

[89]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[90]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[91]  R. Burton,et al.  Density and uniqueness in percolation , 1989 .

[92]  P. Mathieu,et al.  Quenched Invariance Principles for Random Walks with Random Conductances , 2006, math/0611613.

[93]  Codina Cotar,et al.  Decay of covariances, uniqueness of ergodic component and scaling limit for a class of $\nabla\phi$ systems with non-convex potential , 2008, 0807.2621.

[94]  J. Deuschel,et al.  On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to ∇ϕ interface model , 2005 .

[95]  O. Schramm,et al.  Contour lines of the two-dimensional discrete Gaussian free field , 2006, math/0605337.

[96]  J. Černý,et al.  Convergence to fractional kinetics for random walks associated with unbounded conductances , 2011 .

[97]  Jeppe C. Dyre,et al.  Universality of ac conduction in disordered solids , 2000 .

[98]  J. Nash.  Continuity of Solutions of Parabolic and Elliptic Equations , 1958 .

[99]  M. Biskup,et al.  Functional CLT for Random Walk Among Bounded Random Conductances , 2007, math/0701248.

[100]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[101]  Thierry Delmotte,et al.  Parabolic Harnack inequality and estimates of Markov chains on graphs , 1999 .

[102]  Erwin Bolthausen,et al.  Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field , 2010, 1005.5417.

[103]  Nicholas T. Varopoulos,et al.  Analysis and Geometry on Groups , 1993 .

[104]  S. Olla,et al.  Equilibrium Fluctuations for $\nabla_{\varphi}$ Interface Model , 2001 .

[105]  Geoffrey Grimmett.  The Random-Cluster Model , 2002, math/0205237.

[106]  D. Aronson,et al.  Bounds for the fundamental solution of a parabolic equation , 1967 .

[107]  Michael Lin,et al.  The central limit theorem for Markov chains started at a point , 2003 .

[108]  Ofer Zeitouni,et al.  Quenched invariance principle for random walks in balanced random environment , 2010, 1003.3494.

[109]  Mark Jerrum,et al.  Conductance and the Rapid Mixing Property for Markov Chains: the Approximation of the Permanent Resolved (Preliminary Version) , 1988, STOC 1988.

[110]  Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards , 2010, 1009.0048.

[111]  Codina Cotar,et al.  Decay of covariances , uniqueness of ergodic component and scaling limit for a class of ∇ φ systems with non-convex potential , 2008 .

[112]  S. Popov,et al.  Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances , 2010, 1011.1196.

[113]  M. Biskup,et al.  Trapping in the Random Conductance Model , 2012, 1202.2587.

[114]  Gady Kozma,et al.  Disorder, entropy and harmonic functions , 2011, 1111.4853.

[115]  D. Stroock,et al.  Upper bounds for symmetric Markov transition functions , 1986 .

[116]  Volker Strassen,et al.  Almost sure behavior of sums of independent random variables and martingales , 1967 .

[117]  Elchanan Mossel,et al.  On the mixing time of a simple random walk on the super critical percolation cluster , 2000 .

[118]  A conditional quenched CLT for random walks among random conductances on $\mathbb{Z}^d$ , 2011, 1108.5616.

[119]  László Lovász,et al.  Faster mixing via average conductance , 1999, STOC '99.

[120]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[121]  J. Doob.  Stochastic processes , 1953 .

[122]  J. Deuschel,et al.  Invariance principle for the random conductance model with unbounded conductances. , 2010, 1001.4702.

[123]  Finite volume approximation of the effective diffusion matrix: The case of independent bond disorder , 2001, math/0110215.

[124]  Scott Schumacher,et al.  Diffusions with random coefficients , 1984 .

[125]  J. Deuschel,et al.  Invariance principle for the random conductance model , 2013 .

[126]  M. Rosenblatt.  Central limit theorem for stationary processes , 1972 .

[127]  T. Spencer,et al.  On homogenization and scaling limit of some gradient perturbations of a massless free field , 1997 .

[128]  Yu Zhang,et al.  Random walk on the infinite cluster of the percolation model , 1993 .

[129]  O. Boukhadra,et al.  Heat-kernel estimates for random walk among random conductances with heavy tail , 2008, 0812.2669.

[130]  H. Poincaré,et al.  Percolation ? , 1982 .

[131]  H. Spohn,et al.  Motion by Mean Curvature from the Ginzburg-Landau Interface Model , 1997 .

[132]  S. V. Fomin,et al.  Ergodic Theory , 1982 .

[133]  Gabor Pete.  A note on percolation on $Z^d$: isoperimetric profile via exponential cluster repulsion , 2007 .

[134]  J. Černý.  On Two-Dimensional Random Walk Among Heavy-Tailed Conductances , 2011 .