Interfaces in planar Ising and Potts models a review Yvan V elenik Université de Genève
Definition of the models 1/21
Ising and Potts models Box: Λ ⋐ Z 2 ◮ Boundary condition: η ∈ { 1 , . . . , q } ∂ Λ ◮ 2/21
Ising and Potts models Box: Λ ⋐ Z 2 ◮ Boundary condition: η ∈ { 1 , . . . , q } ∂ Λ ◮ Configurations in Λ : ◮ Ω Λ = { 1 , . . . , q } Λ The Ising model corresponds to q = 2 2/21
Ising and Potts models Box: Λ ⋐ Z 2 ◮ Boundary condition: η ∈ { 1 , . . . , q } ∂ Λ ◮ Configurations in Λ : ◮ Ω Λ = { 1 , . . . , q } Λ The Ising model corresponds to q = 2 Energy of σ ∈ Ω Λ with b.c. η : ◮ � � H Λ; η ( σ ) = 1 { σ i � = σ j } + 1 { σ i � = η j } i , j ∈ Λ i ∈ Λ , j ∈ ∂ Λ i ∼ j i ∼ j 2/21
Ising and Potts models Box: Λ ⋐ Z 2 ◮ Boundary condition: η ∈ { 1 , . . . , q } ∂ Λ ◮ Configurations in Λ : ◮ Ω Λ = { 1 , . . . , q } Λ The Ising model corresponds to q = 2 Energy of σ ∈ Ω Λ with b.c. η : ◮ � � H Λ; η ( σ ) = 1 { σ i � = σ j } + 1 { σ i � = η j } i , j ∈ Λ i ∈ Λ , j ∈ ∂ Λ i ∼ j i ∼ j Gibbs measure in Λ with boundary condition η , at inverse temperature β ≥ 0 : ◮ 1 µ η e − β H Λ; η ( σ ) Λ; β ( σ ) = Z η Λ; β σ ∈ Ω Λ e − β H Λ; η ( σ ) is the partition function where Z η Λ; β = � Measures with constant b.c.: µ k Λ; β = µ η k Λ; β , where 1 ≤ k ≤ q and η k ≡ k ◮ 2/21
Ising and Potts models Let β c = log( 1 + √ q ) be the critical inverse temperature . Typical configurations under µ 1 Λ; β : β < β c β > β c In the sequel: we always assume that β > β c . 3/21
Definition of the interface 4/21
Properties of the interface: definition Consider the Potts model in a box with Dobrushin boundary condition : u 5/21
Properties of the interface: definition Consider the Potts model in a box with Dobrushin boundary condition : u 5/21
Properties of the interface: definition Consider the Potts model in a box with Dobrushin boundary condition : We are interested in the behavior of the interface (the set of purple edges). 5/21
Properties of the interface: some milestones Profile of expected magnetization Abraham, Reed 1976: u = e 2 , Ising, β > β c ◮ Abraham, Upton 1988: arbitrary u , Ising, β > β c ◮ Microscopic structure Bricmont, Lebowitz, Pfister 1981: u = e 2 , Ising, β ≫ 1 ◮ Campanino, Ioffe, V. 2003: arbitrary u , Ising, β > β c ◮ Campanino, Ioffe, V. 2008: arbitrary u , Ising, β > β c ◮ Fluctuations Gallavotti 1972: order n 1 / 2 , u = e 2 , Ising, β ≫ 1 ◮ Higuchi 1979: invariance principle, u = e 2 , Ising, β ≫ 1 ◮ Greenberg, Ioffe 2005: invariance principle, arbitrary u , Ising, β > β c ◮ Campanino, Ioffe, V. 2008: invariance principle, arbitrary u , Potts, β > β c ◮ 6/21
Properties of the interface: profile of expected magnetization Roughly speaking, explicit computation of the profile e 2 i �→ µ Λ n ; β ( σ i = 1 ) for the planar Ising model. The “transition region” has width O ( √ n ) . 7/21
Properties of the interface: profile of expected magnetization Roughly speaking, explicit computation of the profile e 2 i �→ µ Λ n ; β ( σ i = 1 ) for the planar Ising model. The “transition region” has width O ( √ n ) . PRO CON explicit expressions, including requires integrability constants provides little understanding no info on typical configurations no direct access to interface 7/21
Properties of the interface: profile of expected magnetization Roughly speaking, explicit computation of the profile e 2 i �→ µ Λ n ; β ( σ i = 1 ) for the planar Ising model. The “transition region” has width O ( √ n ) . In particular, results compatible with various scenarios , including: “fat” interface, of width ∼ √ n ◮ “string-like” interface, exhibiting Gaussian fluctuations with variance ∼ n ◮ 7/21
Properties of the interface: microscopic structure 8/21
Properties of the interface: microscopic structure Ornstein–Zernike theory (Campanino, Ioffe, V. 2003, 2008 and Ott, V., 2018): concatenation of independent microscopic pieces (exponential tails) ◮ � interface has bounded average width 8/21
Properties of the interface: microscopic structure Ornstein–Zernike theory (Campanino, Ioffe, V. 2003, 2008 and Ott, V., 2018): concatenation of independent microscopic pieces (exponential tails) ◮ � interface has bounded average width strong form of coupling with a directed random walk ◮ � enables detailed analysis of fluctuations 8/21
Parenthesis: regularity properties of the Wulff shape 9/21
Parenthesis: regularity properties of the Wulff shape The Wulff (equilibrium crystal) shape describes the (deterministic) shape of a macroscopic droplet of one stable phase immersed in another stable phase. 10/21
Parenthesis: regularity properties of the Wulff shape The Wulff (equilibrium crystal) shape describes the (deterministic) shape of a macroscopic droplet of one stable phase immersed in another stable phase. The emergence of the Wulff shape in the continuum limit of 2d systems with fixed “magnetization” has been understood since the 1990s (Dobrushin, Kotecký, Shlosman, Pfister, Ioffe, Schonmann, V., ...). 10/21
Parenthesis: regularity properties of the Wulff shape Theorem (Campanino, Ioffe, V. 2003, 2008) The boundary of the Wulff shape of the Potts model on Z 2 at any β > β c is analytic and strictly convex , with an everywhere positive curvature . 11/21
Parenthesis: regularity properties of the Wulff shape Theorem (Campanino, Ioffe, V. 2003, 2008) The boundary of the Wulff shape of the Potts model on Z 2 at any β > β c is analytic and strictly convex , with an everywhere positive curvature . In particular, the Wulff shape of the 2d Potts model has no facet, at any positive temperature � no roughening transition in the 2 d Potts model. 11/21
Back to interface fluctuations 12/21
Properties of the interface: fluctuations Theorem (Greenberg, Ioffe 2005, Campanino, Ioffe, V. 2008) Let u be a unit vector in R 2 and β > β c . The interface of the 2 d Potts model in the direction u weakly converges, under diffusive scaling, to the distribution of √ χ β B t where B t is the standard Brownian bridge on [ 0 , 1 ] and χ β is the curvature of the Wulff shape at the unique point t of its boundary where the normal is u . u t Wulff shape 13/21
Pinning of the interface 14/21
Pinning of the interface 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 n row of modified coupling constants (purple), with value J ≥ 0 instead of 1 ◮ 15/21
Pinning of the interface 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 n row of modified coupling constants (purple), with value J ≥ 0 instead of 1 ◮ first considered by Abraham in 1981 for the 2 d Ising model (exact computations) ◮ 15/21
Pinning of the interface Consider the above setting for an arbitrary Potts model on Z 2 . Theorem (Ott, V. 2018) The interface is localized (fluctuations have bounded variance) for all J < 1 . J = 1 J = 1 2 16/21
Entropic repulsion and critical prewetting 17/21
Entropic repulsion & critical prewetting We consider an Ising model in a square box of sidelength n , with the following boundary condition: 18/21
Entropic repulsion & critical prewetting We consider an Ising model in a square box of sidelength n , with the following boundary condition: Theorem (Ioffe, Ott, V., Wachtel 2019) The interface weakly converges, under diffusive scaling, to √ χ β e t , where e t is the standard Brownian excursion. (This holds for general Potts models.) 18/21
Entropic repulsion & critical prewetting We consider an Ising model in a square box of sidelength n , with the following boundary condition: Introduce now a magnetic field h>0 favoring blue spins 18/21
Entropic repulsion & critical prewetting We consider an Ising model in a square box of sidelength n , with the following boundary condition: Introduce now a magnetic field h>0 favoring blue spins ◮ � yellow phase becomes thermodynamically unstable ◮ � layer becomes microscopic 18/21
Entropic repulsion & critical prewetting However, the width of this layer increases as h ↓ 0: 19/21
Entropic repulsion & critical prewetting However, the width of this layer increases as h ↓ 0: We are interested in the scaling limit of this layer as n → ∞ and h ↓ 0. A good choice is to set h = h ( n ) = λ n for some λ > 0. 19/21
Entropic repulsion & critical prewetting Scale the interface by n − 1 / 3 vertically, n − 2 / 3 horizontally ◮ Let m ∗ β = � σ 0 � + β = denote the spontaneous magnetization ◮ Let ◮ 1 / 3 r + ω 1 ( 4 m ∗ ϕ 0 ( r ) = Ai � � β /χ β ) where Ai is the Airy function and ω 1 its first zero. 20/21
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