Lectures on the spin and loop O(n) models

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The classical spin O(n) model is a model on a d-dimensional lattice in which a vector on the (n-1) -dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model (n=1), the XY model (n=2) and the Heisenberg model (n=3). We discuss questions of long-range order and decay of correlations in the spin O(n) model for different combinations of the lattice dimension d and the number of spin components n. The loop O(n) model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight n≥0 and an edge weight (n=0). Special cases include self-avoiding walk (n=1), the Ising model (n=x=1), critical percolation (formula presented), dimer model (formula presented), proper 4-coloring (formula presented), integer-valued (n=2) and tree-valued (integer (formula presented)) Lipschitz functions and the hard hexagon model (formula presented). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin O(n) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.

Original languageEnglish
Title of host publicationSojourns in Probability Theory and Statistical Physics - I - Spin Glasses and Statistical Mechanics, A Festschrift for Charles M. Newman
EditorsVladas Sidoravicius
Number of pages75
ISBN (Print)9789811502934
StatePublished - 2019
EventInternational Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China
Duration: 25 Mar 201627 Mar 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceInternational Conference on Probability Theory and Statistical Physics, 2016


  • Berezinskii–Kosterlitz–Thouless transition
  • Chessboard estimate
  • Conformal loop ensemble
  • Critical percolation on the triangular lattice
  • Decay of correlations
  • Dilute potts model
  • Dimer model
  • Gaussian domination
  • Graphical representation
  • Hard hexagon model
  • Heisenberg model
  • Infra-red bound
  • Ising model
  • Lipschitz functions
  • Loop O(n) model
  • Macroscopic loops
  • Mermin–Wagner
  • Microscopic loops
  • Phase transitions
  • Proper 4-coloring of the triangular lattice
  • Reflection positivity
  • Schramm–Loewner evolution
  • Self-avoiding walk
  • Spin O(n) model
  • Spontaneous magnetization
  • Symmetry breaking
  • XY model


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