V5F5 - Advanced Topics in Applied Probability:

Introduction to Lattice Models

Course announcement

WS20/21: Tuesdays 10:15–12:00 and Thursdays 10:15–12:00.  

Online: zoom link can be found in eCampus

If you are unable to find the zoom link, please contact me:

firstname.lastname @


We present examples in the zoo of critical lattice models, originating from statistical physics, now very popular also in mathematics. No physics background is needed – this serves only as motivation.

We discuss the meaning of phase transitions, via example models such as variants of percolation, Ising model, polymers, and O(n)-models. We outline some recent results on the mathematical understanding of critical phenomena. We will see examples of applications of discrete complex analysis, which has lead to recent breakthrough results, and renormalization group techniques, originating from physics ideas but now finding their use in mathematics as well. Some Gaussian fields will be discussed towards the end of the course.

The precise plan can also depend on the participants’ wishes. If you have any questions or wishes, please don't hesitate to contact me.



Lecture sketches are here



  • Set 1 (graphs, contructing current using random walk, network simplifications) Notes here
  • Set 2 (harmonic functions, their relation to random walk and resistance, Green's function, discrete Laplacian) Notes here
  • Set 3 (recurrence/transience, quasi-isometries, weak limits) Notes here
  • Set 4 (basics on percolation, Matrix-tree theorem, MC-tree theorem) Notes here
  • Set 5 (basics on random-cluster model: Markov property, FKG, etc; and Holley's inequality as a useful tool) Notes here
  • Set 6 (more properties of random-cluster model and Ising model) Notes here

The exercises are highly recommended. Some of them you might find familiar already. Slightly harder ones are marked with a star and these are optional.


Course Log:

Past and future (planned) topics of lectures, as well as some literary references, will appear here.

  • 27.10: introduction and warm-up (link to slides in eCampus)
  • 29.10: basic concepts (graphs, lattices, duality, Euler's formula), random walks on graphs, (discrete-time) reversible Markov chains (ref. [G, Chapter 1], see also [B: Chapter I])
  • 3.11: discrete harmonic functions, electrical networks, Kirchoff and Ohm laws (ref. [G, Chapter 1], see also Exercises 2)
  • 5.11: construction of current via spanning trees, conceps of energy, effective resistance, Thomson variational principle and Rayleigh's monotonicity principle (ref. [G, Chapter 1])
  • 10.11: Characterizing recurrence/transience of RW on graphs (ref. [G, Chapter 1])
  • 12.11: Polya's theorem: robust proof (ref. [G, Chapter 1])
  • 17.11: uniform spanning tree, negative association, thermodynamic limit (ref. [G, Chapter 2], see also [LP: Chapter 10.1])
  • 19.11: thermodynamic limit, stochastic domination of free/wired UST, Wilson's algorithm, loop-erased walk (ref. [G, Chapter 2], see also [LP: Chapter 10.2])
  • 24.11: Wilson's algorithm, LERW, Markov chain - Tree theorem (ref. [G, Chapter 2], see also [LP: Chapter 4])
  • 26.11: discussion on Exercise Sets 1-2
  • 01.12: scaling limits of LERW, UST Peano curve; crash course on SLE curves (ref. [G, Chapter 2.5])
  • 03.12: percolation, phase transition, basic properties (ref. parts of [G, Chapters 3.1, 3.4, 4.7]; see also Chapters 1-2 of Grimmett's book "Percolation")
  • 08.12: proof of phase transition and criticality in percolation (ref. [G, Chapter 3.1])
  • 10.12: Burton-Keane argument (ref. [G, Chapter 5.3])
  • 15.12: exponential decay in subcritical regime (ref. [G, Chapter 5.1])
  • 17.12: mean-field bound in supercritical regime, critical exponents, \theta(p_c)=0 for Z^2 (ref. [G, Chapters 5.1, 5.4, 5.6])
  • 22.12: p_c=1/2 for Z^2, Cardy's crossing probability formula and conformal invariance on the triangular lattice (ref. [G, Chapters 5.6, 5.7])
  • 24.12: Merry Christmas!
  • 07.01: discussion on Exercise Sets 3-4
  • 12.01: Random-cluster model, basic properties, phase transition (ref. [G, Chapters 8.3, 8.2]; see also [D, Chapter 4])
  • 14.01: uniqueness for thermodynamic limit of Random-cluster model, free energy (ref. [D, Theorem 4.30 in Chapter 4.5]; see also [G, Theorem 8.18 in Chapter 8.3])
  • 19.01: main properties of Random-cluster model (ref. [D, Chapter 4 (and 5)]), Ising & Potts model, Edwards-Sokal coupling (ref. [D, Chapter 7.3]; see also [G, Chapter 8.1])
  • 21.01: consequences of Edwards-Sokal coupling, properties of Ising model (ref. [D, Chapter 7]; see also [G, Chapter 8.1] and [FV, Chapter 3])
  • 26.01: phase transition of Ising model and uniqueness / non-uniqueness of thermodynamic limit (ref. [D, Chapter 7.4])
  • 28.01: no lecture
  • 02.02: some details needed in Peierls argument for Ising model: high- and low temperature expansions (ref. [D, Chapter 7.5])
  • 04.02: discussion on Exercise Set 5 (maybe 1 hour), Peierls argument for Ising model (ref. [D, Chapter 7.5.3])
  • 09.02: Spin and loop-O(n) models, BKT type phase transition (ref. [D, Chapter 12.3-12.4])
  • 11.02: discussion on Exercise Set 6 (maybe 1 hour), how are spin and loop-O(n) models related, Nienhuis's conjecture about phase transition (ref. [D, Chapter 12.3-12.4])



The main sources are parts of these lecture notes / books:

[G]: G. Grimmett: Probability on Graphs, <>

[D]: H. Duminil-Copin: Parafermionic observables and their applications to planar statistical physics models, <>

[LP]: R. Lyons & Y. Peres: Probability on Trees and Networks,<>

Other useful textbooks:

[FV]: S. Friedli & Y. Velenik: Statistical Mechanics of Lattice Systems,<>

Grimmett's book "Percolation"

Grimmett's book "Random-cluster model"

Additional material and advanced topics:

[B]: B. Bollobas: Modern Graph Theory, <>

[BBS]: R. Bauerschmidt, D. Brydges & G. Slade: Introduction to a renormalisation group method, <>

[BDGS]: R. Bauerschmidt, H. Duminil-Copin, J. Goodman & G. Slade: Lectures on Self-Avoiding Walks,<>