The self-avoiding walk on the hexagonal lattice Hugo Duminil-Copin - PowerPoint PPT Presentation

The self-avoiding walk on the hexagonal lattice Hugo Duminil-Copin Universit´ e de Gen` eve Stanislav Smirnov Universit´ e de Gen` eve & St. Petersburg State University January 2011 Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Self-Avoiding Walks on the hexagonal lattice H : a Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Self-Avoiding Walks on the hexagonal lattice H : a Conjecture (Flory, 1948; Nienhuis, 1982) Precise asymptotics for the mean-square displacement c n of SAWs of length n : � | ω ( n ) | 2 � ∼ Dn 2 ν as n − → ∞ , where ν := 3 / 4 . Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Self-Avoiding Walks on the hexagonal lattice H : a Conjecture (Flory, 1948; Nienhuis, 1982) Precise asymptotics for the mean-square displacement and for the number c n of SAWs of length n : � | ω ( n ) | 2 � ∼ Dn 2 ν as n − → ∞ , c n ∼ An γ − 1 µ n as n − → ∞ c √ � where ν := 3 / 4 and µ c := 2 + 2, γ := 43 / 32. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Self-Avoiding Walks on the hexagonal lattice H : a Conjecture (Flory, 1948; Nienhuis, 1982) Precise asymptotics for the mean-square displacement and for the number c n of SAWs of length n : � | ω ( n ) | 2 � ∼ Dn 2 ν as n − → ∞ , c n ∼ An γ − 1 µ n as n − → ∞ c √ � where ν := 3 / 4 and µ c := 2 + 2, γ := 43 / 32. γ and ν are universal; µ c is lattice-dependent. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Theorem (H. Duminil-Copin, S. Smirnov, 2010) √ 1 � The connective constant satisfies µ c := lim n →∞ c = 2 + 2. n n Easy observations: 1 c n + m < c n · c m ⇒ ∃ µ c := lim n →∞ c n , n √ 2 n / 2 ≤ c n ≤ 3 · 2 n − 1 ⇒ 2 ≤ µ c ≤ 2 . Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Theorem (H. Duminil-Copin, S. Smirnov, 2010) √ 1 � The connective constant satisfies µ c := lim n →∞ c = 2 + 2. n n Easy observations: 1 c n + m < c n · c m ⇒ ∃ µ c := lim n →∞ c n , n √ 2 n / 2 ≤ c n ≤ 3 · 2 n − 1 ⇒ 2 ≤ µ c ≤ 2 . The generating function (diverges µ < µ c , converges µ > µ c ): µ − ℓ ( ω ) = � � c n · µ − n . G ( µ ) := ω n Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Theorem (H. Duminil-Copin, S. Smirnov, 2010) √ 1 � The connective constant satisfies µ c := lim n →∞ c = 2 + 2. n n Easy observations: 1 c n + m < c n · c m ⇒ ∃ µ c := lim n →∞ c n , n √ 2 n / 2 ≤ c n ≤ 3 · 2 n − 1 ⇒ 2 ≤ µ c ≤ 2 . The generating function (diverges µ < µ c , converges µ > µ c ): µ − ℓ ( ω ) = � � c n · µ − n . G ( µ ) := ω n It is expected that G ( µ ) ∼ ( µ c − µ ) − γ . Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Theorem (H. Duminil-Copin, S. Smirnov, 2010) √ 1 � The connective constant satisfies µ c := lim n →∞ c = 2 + 2. n n Easy observations: 1 c n + m < c n · c m ⇒ ∃ µ c := lim n →∞ c n , n √ 2 n / 2 ≤ c n ≤ 3 · 2 n − 1 ⇒ 2 ≤ µ c ≤ 2 . The generating function (diverges µ < µ c , converges µ > µ c ): µ − ℓ ( ω ) = � � c n , a → z · µ − n . G a → z ( µ ) := ω ⊂ Ω: a → z n It is expected that G ( µ ) ∼ ( µ c − µ ) − γ . Try to count simpler objects, bridges : Walks that never go below the first step and above the last one. The number of bridges grows at the same (exponential) speed as walks. a Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition A self-avoiding bridge is a SAW ω such that the first site is of minimal second coordinate and the last one of maximal second coordinate. Let b n be the number of self-avoiding bridges of length n . 0 Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition A self-avoiding bridge is a SAW ω such that the first site is of minimal second coordinate and the last one of maximal second coordinate. Let b n be the number of self-avoiding bridges of length n . 0 Proposition (Hammersley 1961) µ c is the same for bottom-top bridges, bottom-bottom bridges, loops. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition A self-avoiding bridge is a SAW ω such that the first site is of minimal second coordinate and the last one of maximal second coordinate. Let b n be the number of self-avoiding bridges of length n . 0 Proposition (Hammersley 1961) µ c is the same for bottom-top bridges, bottom-bottom bridges, loops. γ is expected to be different: 9 / 16, 9 / 16, − 1 / 2. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition A self-avoiding bridge is a SAW ω such that the first site is of minimal second coordinate and the last one of maximal second coordinate. Let b n be the number of self-avoiding bridges of length n . 0 Proposition (Hammersley 1961) µ c is the same for bottom-top bridges, bottom-bottom bridges, loops. γ is expected to be different: 9 / 16, 9 / 16, − 1 / 2. b n ≤ c n for obvious reasons. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition A self-avoiding bridge is a SAW ω such that the first site is of minimal second coordinate and the last one of maximal second coordinate. Let b n be the number of self-avoiding bridges of length n . 0 Proposition (Hammersley 1961) µ c is the same for bottom-top bridges, bottom-bottom bridges, loops. γ is expected to be different: 9 / 16, 9 / 16, − 1 / 2. b n ≤ c n for obvious reasons. Moreover, c n ≤ r 2 n b n where r n is the number of partitions of n into increasing positive integers. Since r n ≤ Ce c √ n , we obtain that b n and c n are logarithmically equivalent. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition The winding W ω ( a , b ) of a curve ω between a and b is the rotation (in radians) of the curve between a and b . b b a a W γ ( a, b ) = 0 W γ ( a, b ) = 2 π Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition The winding W ω ( a , b ) of a curve ω between a and b is the rotation (in radians) of the curve between a and b . b b a a W γ ( a, b ) = 0 W γ ( a, b ) = 2 π With this definition, we can define the parafermionic operator for a ∈ ∂ Ω and z ∈ Ω: � e − i σ W ω ( a , z ) µ − ℓ ( ω ) . F ( z ) = F ( a , z , µ, σ ) := ω ⊂ Ω: a → z Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition The winding W ω ( a , b ) of a curve ω between a and b is the rotation (in radians) of the curve between a and b . b b a a W γ ( a, b ) = 0 W γ ( a, b ) = 2 π With this definition, we can define the parafermionic operator for a ∈ ∂ Ω and z ∈ Ω: � e − i σ W ω ( a , z ) µ − ℓ ( ω ) . F ( z ) = F ( a , z , µ, σ ) := ω ⊂ Ω: a → z Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

Definition The winding W ω ( a , b ) of a curve ω between a and b is the rotation (in radians) of the curve between a and b . b b a a W γ ( a, b ) = 0 W γ ( a, b ) = 2 π With this definition, we can define the parafermionic operator for a ∈ ∂ Ω and z ∈ Ω: � e − i σ W ω ( a , z ) µ − ℓ ( ω ) . F ( z ) = F ( a , z , µ, σ ) := ω ⊂ Ω: a → z Lemma (Discrete integrals on elementary contours vanish) √ � 2 and σ = 5 If µ = µ ∗ = 2 + 8 , then F satisfies the following relation for every vertex v ∈ V (Ω) , ( p − v ) F ( p ) + ( q − v ) F ( q ) + ( r − v ) F ( r ) = 0 where p , q , r are the mid-edges of the three edges adjacent to v. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

We write c ( ω ) for the contribution of the walk ω to the sum. Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

We write c ( ω ) for the contribution of the walk ω to the sum. One can partition the set of walks ω finishing at p , q or r into pairs and triplets of walks: γ 1 γ 2 γ 1 γ 2 γ 3 Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice

We write c ( ω ) for the contribution of the walk ω to the sum. One can partition the set of walks ω finishing at p , q or r into pairs and triplets of walks: γ 1 γ 2 γ 1 γ 2 γ 3 In the first case, c ( ω 1 ) + c ( ω 2 ) = ( q − v ) e − i σ W ω 1 ( a , q ) µ − ℓ ( ω 1 ) + ( r − v ) e − i σ W ω 2 ( a , r ) µ − ℓ ( ω 2 ) = ( p − v ) e − i σ W ω 1 ( a , p ) µ − ℓ ( ω 1 ) � 3 e − i σ · − 4 π � e i 2 π + e − i 2 π 3 e − i σ · 4 π 3 3 Hugo Duminil-Copin & Stanislav Smirnov The self-avoiding walk on the hexagonal lattice