Order - disorder operators in planar and almost planar graphs (2) Hugo Duminil-Copin, I.H.´ E.S. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
The main statement Consider the Ising’s Hamiltonian (with free boundary conditions) on a finite subgraph G of the square lattice Z 2 with coupling constants J xy ≥ 0, � def H G ( σ ) = − J xy σ x σ y x , y ∈ G and the associated measure at inverse-temperature β defined for any f , � f ( σ ) exp[ − β H G ( σ )] σ ∈{± 1 } G � f � G ,β = � . exp[ − β H G ( σ )] σ ∈{± 1 } G Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
The main statement Consider the Ising’s Hamiltonian (with free boundary conditions) on a finite subgraph G of the square lattice Z 2 with coupling constants J xy ≥ 0, � def H G ( σ ) = − J xy σ x σ y x , y ∈ G and the associated measure at inverse-temperature β defined for any f , � f ( σ ) exp[ − β H G ( σ )] σ ∈{± 1 } G � f � G ,β = � . exp[ − β H G ( σ )] σ ∈{± 1 } G Focus on symmetric finite range (i.e. J xy = 0 for | x − y | ≥ R ) Ising model at the critical inverse temperature β c on the upper half-plane H . Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
The main statement Consider the Ising’s Hamiltonian (with free boundary conditions) on a finite subgraph G of the square lattice Z 2 with coupling constants J xy ≥ 0, � def H G ( σ ) = − J xy σ x σ y x , y ∈ G and the associated measure at inverse-temperature β defined for any f , � f ( σ ) exp[ − β H G ( σ )] σ ∈{± 1 } G � f � G ,β = � . exp[ − β H G ( σ )] σ ∈{± 1 } G Focus on symmetric finite range (i.e. J xy = 0 for | x − y | ≥ R ) Ising model at the critical inverse temperature β c on the upper half-plane H . Theorem (Aizenman, Duminil-Copin, Tassion, Warzel (2016)) For x 1 , . . . , x 2 n found in this order on the boundary of H , �� � � � σ x 1 · · · σ x 2 n � H ,β c ∼ Pfaff � σ x i σ x j � H ,β c 1 ≤ i < j ≤ 2 n as min | x i − x j | tends to infinity. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
Part I. Planar Case Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents ) m = ( m xy : x , y ∈ G ). Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
Part I. Planar Case Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents ) m = ( m xy : x , y ∈ G ). As observed by Griffiths, Hurst and Sherman (1970), the identity ∞ � ( β J xy σ x σ y ) m xy exp[ β J xy σ x σ y ] = m xy ! m xy =0 allows us to write for σ A = � x ∈ A σ x , � � � def σ A exp[ − β H G ( σ )] = σ A exp[ β J xy σ x σ y ] x , y ∈ G σ ∈{± 1 } G σ ∈{± 1 } G Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
Part I. Planar Case Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents ) m = ( m xy : x , y ∈ G ). As observed by Griffiths, Hurst and Sherman (1970), the identity ∞ � ( β J xy σ x σ y ) m xy exp[ β J xy σ x σ y ] = m xy ! m xy =0 allows us to write for σ A = � x ∈ A σ x , � � � � switch sums σ I [ x ∈ A ]+∆( m , x ) σ A exp[ − β H G ( σ )] = w ( m ) x x ∈ G σ ∈{± 1 } G m σ ∈{± 1 } G = � β m xy = � def def where w ( m ) m xy ! and ∆( m , x ) y m xy . x ∼ y Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
Part I. Planar Case Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents ) m = ( m xy : x , y ∈ G ). As observed by Griffiths, Hurst and Sherman (1970), the identity ∞ � ( β J xy σ x σ y ) m xy exp[ β J xy σ x σ y ] = m xy ! m xy =0 allows us to write for σ A = � x ∈ A σ x , � 2 | G | � σ A exp[ − β H G ( σ )] = w ( m ) , σ ∈{± 1 } G ∂ m = A = � β m xy def def where w ( m ) m xy ! and sources ∂ m = { x ∈ G , ∆( m , x ) odd } . x ∼ y based on the fact that for each fixed x ∈ G , the map flipping σ x is an involution on spin configurations. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . 0 M 1 1 n 0 0 0 2 2 2 3 b 2 0 a b 0 1 a 1 5 Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
An interpretation of currents in terms of loops Identify m = ( m xy : x , y ∈ G ) with a (multi-)graph M with m xy edges between x and y . M b a A current m with sources ∂ m = A can be seen as a collection of loops together with paths pairing the vertices of A together. Hugo Duminil-Copin, I.H.´ Order - disorder operators in planar and almost planar graphs (2) E.S.
Recommend
More recommend
