Lectures on the Spin and Loop O(n) Models

DOI10.1007/978-981-15-0294-1_10zbMath1446.82019arXiv1708.00058OpenAlexW2740322853MaRDI QIDQ3297366

Ron Peled, Yinon Spinka

Publication date: 3 July 2020

Published in: Springer Proceedings in Mathematics & Statistics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1708.00058

zbMATH Keywords

phase transitionsIsing modelchessboard estimatereflection positivityHeisenberg modelBerezinskii-Kosterlitz-Thouless transitionsymmetry breakingLipschitz functionsspontaneous magnetizationself-avoiding walkdecay of correlationsdimer modelSchramm-Loewner evolutionconformal loop ensembleXY modelgraphical representationloop \(\mathrm{O}(n)\) modelhard hexagon modeldilute Potts modelGaussian dominationinfra-red boundmacroscopic loopscritical percolation on the triangular latticeMermin-Wagnermicroscopic loopsproper 4-coloring of the triangular latticespin \(\mathrm{O}(n)\) model

Mathematics Subject Classification ID

Symmetry breaking in quantum theory (81R40) Phase transitions (general) in equilibrium statistical mechanics (82B26) Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics (82B41) Percolation (82B43) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20) Critical phenomena in equilibrium statistical mechanics (82B27) Statistical mechanics of magnetic materials (82D40) Statistical thermodynamics (82B30) Stochastic (Schramm-)Loewner evolution (SLE) (60J67)

Related Items (20)

Finitary codings for the random-cluster model and other infinite-range monotone models ⋮ Lattice gauge theory and a random-medium Ising model ⋮ Uniform Lipschitz functions on the triangular lattice have logarithmic variations ⋮ Macroscopic loops in the loop \(O(n)\) model at Nienhuis' critical point ⋮ Site monotonicity and uniform positivity for interacting random walks and the spin \(O(N)\) model with arbitrary \(N\) ⋮ Variational principle for weakly dependent random fields ⋮ Finitary codings for gradient models and a new graphical representation for the six‐vertex model ⋮ Macroscopic loops in the Bose gas, spin \(O(N)\) and related models ⋮ On the transition between the disordered and antiferroelectric phases of the 6-vertex model ⋮ The critical \(O(N)\) CFT: methods and conformal data ⋮ Macroscopic loops in the \(3 d\) double-dimer model ⋮ An elementary proof of phase transition in the planar XY model ⋮ Shifted critical threshold in the loop \(O(n)\) model at arbitrarily small \(n\) ⋮ Strongly correlated random interacting processes. Abstracts from the workshop held January 28 -- February 3, 2018 ⋮ Exponential decay in the loop \(O(n)\) model on the hexagonal lattice for \(n > 1\) and \(x<\frac{1}{\sqrt{3}}+\varepsilon (n)\) ⋮ Site-monotonicity properties for reflection positive measures with applications to quantum spin systems ⋮ Exponential decay of transverse correlations for \(O(N)\) spin systems and related models ⋮ Loop correlations in random wire models ⋮ Infinite DLR measures and volume-type phase transitions on countable Markov shifts ⋮ A general method for lower bounds on fluctuations of random variables

Cites Work

• Correlation inequalities for the quantum XY model
• Two-dimensional critical percolation: the full scaling limit
• Methods of contemporary mathematical statistical physics. Lecture notes presented at the 5th Prague summer school on mathematical statistical physics, Prague, Czech Republik, September 2006
• Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
• A low temperature expansion for classical \(N\)-vector models. I: A renormalization group flow
• On the slow decay of \(O(2)\) correlations in the absence of topological excitations: Remark on the Patrascioiu-Seiler model
• Precolation of the minority spins in high-dimensional Ising models
• Random currents and continuity of Ising model's spontaneous magnetization
• The scaling limit of critical Ising interfaces is \(\mathrm{CLE}_3\)
• Counting regions with bounded surface area
• Continuum nonsimple loops and 2D critical percolation
• Critical percolation exploration path and \(\mathrm{SLE}_{6}\): a proof of convergence
• The Spherical Model of a Ferromagnet
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