Extreme boundary conditions and random tilings

Abstract

Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]

arXiv commentExpanded version of the lectures given at the SFT-Paris-2019 school on 'Statistical and Condensed Matter Field Theory'. 66 pages. v2: various minor improvements

References in corpus (48)

• Level-Spacing Distributions and the Airy Kernel
• One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
• Shape Fluctuations and Random Matrices
• Quantum Inverse Scattering Method and Correlation Functions
• Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
• Emergent hydrodynamics in integrable quantum systems out of equilibrium
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Applied Conformal Field Theory
• Quantum quenches in 1+1 dimensional conformal field theories
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• A variational principle for domino tilings
• Growing interfaces uncover universal fluctuations behind scale invariance
• Asymptotics in ASEP with Step Initial Condition
• The one-dimensional KPZ equation and its universality class
• Analytic solution of the Domain Wall non-equilibrium stationary state
• Shannon and entanglement entropies of one- and two-dimensional critical wave functions
• Random Domino Tilings and the Arctic Circle Theorem
• Classical dimers on the triangular lattice
• Inhomogeneous field theory inside the arctic circle
• Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model
• Interacting classical dimers on the square lattice
• Classical dimers with aligning interactions on the square lattice
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Phase transition in the Rényi-Shannon entropy of Luttinger liquids
• The arctic curve of the domain-wall six-vertex model
• Noninteracting fermions in a trap and random matrix theory
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain
• Current fluctuations for anharmonic chains in thermal equilibrium
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
• Entanglement in gapless resonating valence bond states
• Free fermions at the edge of interacting systems
• Onsager and Kaufman's calculation of the spontaneous magnetization of the Ising model
• Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity
• Lectures on the integrability of the 6-vertex model
• Limit shapes for the asymmetric five vertex model
• Onsager and Kaufman's calculation of the spontaneous magnetization of the Ising model: II
• Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model
• Non-integrable dimers: Universal fluctuations of tilted height profiles
• Quantum gravity and the KPZ formula
• Dimers on Rail Yard Graphs
• Inhomogeneous Gaussian Free Field inside the interacting arctic curve
• Rectangular amplitudes, conformal blocks, and applications to loop models
• Haldane relation for interacting dimers
• Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram
• SLE and the free field: Partition functions and couplings
• The propagator of the finite XXZ spin-$\frac{1}{2}$ chain

Cited by in corpus (9)

• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Limit shape phase transitions: a merger of arctic circles
• Exact time evolution formulae in the XXZ spin chain with domain wall initial state
• Double tangent method for two-periodic Aztec diamonds
• Emptiness Formation in Polytropic Quantum Liquids
• Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking
• Gradient variational problems in ${\mathbb{R}}^2$
• Interface fluctuations associated with split Fermi seas
• Emptiness Instanton in Quantum Polytropic Gas