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,
PoissonDirichlet Statistics for the extremes of a logcorrelated Gaussian field.
Electron. J. Probab. 20 (2015), no. 59, 119

M. Biskup, J. Ding and S. Goswami, Return probability and recurrence for the random walk driven by twodimensional Gaussian free field, arXiv:1611.03901

M. Biskup and O. Louidor, Extreme local extrema of twodimensional discrete Gaussian free field, Commun. Math. Phys. 345 (2016), no. 1, 271304

M. Biskup and O. Louidor, Conformal symmetries in the extremal process of twodimensional discrete Gaussian Free Field, arXiv:1410.4676

M. Biskup and O. Louidor, Full extremal process, cluster law and freezing for twodimensional discrete Gaussian Free Field, arXiv:1606.00510

M. Biskup and O. Louidor, On intermediate level sets of twodimensional discrete Gaussian Free Field, arXiv:1612.01424

E. Bolthausen, J.D. Deuschel and G. Giacomin,
Entropic repulsion and the maximum of the twodimensional harmonic crystal,
Ann. Probab. 29 (2001), no. 4, 16701692

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) 114119

M. Bramson,
Maximal displacement of branching Brownian motion.
Commun. Pure Appl. Math. 31 (1978), no. 5, 531581

M. Bramson, J. Ding and O. Zeitouni,
Convergence in law of the maximum of the twodimensional discrete Gaussian free field.
Commun. Pure Appl. Math 69 (2016), no. 1, 62123

M. Bramson and O. Zeitouni,
Tightness of the recentered maximum of the twodimensional discrete Gaussian free field.
Commun. Pure Appl. Math. 65 (2011) 120

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 sinhGordon 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, 48594862

J. Ding,
Exponential and double exponential tails for maximum of twodimensional discrete Gaussian free field,
Probab. Theory Rel. Fields 157 (2013), no. 1, 285299

J. Ding and O. Zeitouni,
Extreme values for twodimensional discrete Gaussian free field, Ann. Probab.
42 (2014) no. 4, 14801515

O. Daviaud,
Extremes of the discrete twodimensional Gaussian free field,
Ann. Probab. 34 (2006) 962986

T.M. Liggett,
Random invariant measures for Markov chains, and independent particle systems,
Z. Wahrsch. Verw. Gebiete 45 (1978) 297313
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 NotesMonograph 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 799801.