Relevant work:
(subject to regular updates)

Discrete GFF:

The course will cover, or otherwise rederive, the content (or at least parts thereof) of the following papers on DGFF or techniques used to study extrema of DGFF:

• L.-P. Arguin and O. Zindy, Poisson-Dirichlet Statistics for the extremes of a log-correlated Gaussian field. Electron. J. Probab. 20 (2015), no. 59, 1--19
• M. Biskup, J. Ding and S. Goswami, Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field, arXiv:1611.03901
• M. Biskup and O. Louidor, Extreme local extrema of two-dimensional discrete Gaussian free field, Commun. Math. Phys. 345 (2016), no. 1, 271--304
• M. Biskup and O. Louidor, Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field, arXiv:1410.4676
• M. Biskup and O. Louidor, Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian Free Field, arXiv:1606.00510
• M. Biskup and O. Louidor, On intermediate level sets of two-dimensional discrete Gaussian Free Field, arXiv:1612.01424
• E. Bolthausen, J.-D. Deuschel and G. Giacomin, Entropic repulsion and the maximum of the two-dimensional harmonic crystal, Ann. Probab. 29 (2001), no. 4, 1670--1692
• E. Bolthausen, J.-D. Deuschel, and O. Zeitouni, Recursions and tightness for the maximum of the discrete, two dimensional gaussian free field, Elect. Commun. Probab. 16 (2011) 114--119
• M. Bramson, Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31 (1978), no. 5, 531--581
• M. Bramson, J. Ding and O. Zeitouni, Convergence in law of the maximum of the two-dimensional discrete Gaussian free field. Commun. Pure Appl. Math 69 (2016), no. 1, 62--123
• M. Bramson and O. Zeitouni, Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Commun. Pure Appl. Math. 65 (2011) 1--20
• D. Carpentier and P. Le Doussal, Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models, Phys. Rev. E 63 (2001) 026110
• H. Castillo and P. Le Doussal, Freezing of dynamical exponents in low dimensional random media. Phys. Rev. Lett. 86 (2001), no. 21, 4859--4862
• J. Ding, Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field, Probab. Theory Rel. Fields 157 (2013), no. 1, 285--299
• J. Ding and O. Zeitouni, Extreme values for two-dimensional discrete Gaussian free field, Ann. Probab. 42 (2014) no. 4, 1480--1515
• O. Daviaud, Extremes of the discrete two-dimensional Gaussian free field, Ann. Probab. 34 (2006) 962--986
• T.M. Liggett, Random invariant measures for Markov chains, and independent particle systems, Z. Wahrsch. Verw. Gebiete 45 (1978) 297--313

General references:

We will also draw on various standard references concerning Gaussian processes and/or random walks:

• R.J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics Lecture Notes--Monograph Series, vol. 12. Institute of Mathematical Statistics, Hayward, CA, 1990, x+160 pp.
• R.J. Adler and J.E. Taylor, Random fields and geometry. Springer Monographs in Mathematics. Springer, New York, 2007.
• G. Choquet and J. Deny. Sur l'equation de convolution μ=μ*σ. C.R. Acad. Sci. Paris (1960) 250 799--801.