Self-avoiding walk on the hexagonal lattice Hugo Duminil-Copin, - PowerPoint PPT Presentation

Self-avoiding walk on the hexagonal lattice Hugo Duminil-Copin, Universit´ e de Gen` eve May 2010 joint work with S. Smirnov Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Consider the hexagonal lattice H , and define c n to be the number of self-avoiding walks of length n starting at the origin. 0 Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Consider the hexagonal lattice H , and define c n to be the number of self-avoiding walks of length n starting at the origin. a We will consider walks starting and ending at mid-edges! Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Consider the hexagonal lattice H , and define c n to be the number of self-avoiding walks of length n starting at the origin. a We will consider walks starting and ending at mid-edges! Combinatorial question: How many such trajectories are there? In other words, what is the value of c n ? Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Theorem (H. D-C, S. Smirnov, 2010) √ 1 � The connective constant µ satisfies µ := lim n →∞ c = 2 + 2 . n n Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Theorem (H. D-C, S. Smirnov, 2010) √ 1 � The connective constant µ satisfies µ := lim n →∞ c = 2 + 2 . n n Strategy : It is sufficient to prove that the partition function Z ( x ) is √ � finite if and only if x > 2 + 2 =: x c , where � x − ℓ ( γ ) Z ( x ) := γ : a → H Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Theorem (H. D-C, S. Smirnov, 2010) √ 1 � The connective constant µ satisfies µ := lim n →∞ c = 2 + 2 . n n Strategy : It is sufficient to prove that the partition function Z ( x ) is √ � finite if and only if x > 2 + 2 =: x c , where � x − ℓ ( γ ) Z ( x ) := γ : a → H We restrict ourself to finite domains Ω and we weight walks by their winding . Ω z mid-edge a Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Definition The winding W Γ ( a , b ) of a curve Γ 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, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Definition The winding W Γ ( a , b ) of a curve Γ 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 γ ( a , z ) x − ℓ ( γ ) . F ( z ) = F ( a , z , x , σ ) := γ ⊂ Ω: a → z Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Definition The winding W Γ ( a , b ) of a curve Γ 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 γ ( a , z ) x − ℓ ( γ ) . F ( z ) = F ( a , z , x , σ ) := γ ⊂ Ω: a → z Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

Definition The winding W Γ ( a , b ) of a curve Γ 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 γ ( a , z ) x − ℓ ( γ ) . F ( z ) = F ( a , z , x , σ ) := γ ⊂ Ω: a → z Lemma (Discrete integrals on elementary contours vanish) If x = x c and σ = 5 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, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We write c ( γ ) for the contribution of the walk γ to the sum. Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We write c ( γ ) for the contribution of the walk γ to the sum. One can partition the set of walks γ finishing at p , q or r into pairs and triplets of walks: γ 1 γ 2 γ 2 γ 1 γ 3 Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We write c ( γ ) for the contribution of the walk γ to the sum. One can partition the set of walks γ finishing at p , q or r into pairs and triplets of walks: γ 1 γ 2 γ 2 γ 1 γ 3 In the first case: c ( γ 1 ) + c ( γ 2 ) = ( q − v ) e − i σ W γ 1 ( a , q ) x − ℓ ( γ 1 ) + ( r − v ) e − i σ W γ 2 ( a , r ) x − ℓ ( γ 2 ) c c � � e i 2 π 3 e − i 5 8 · − 4 π + e − i 2 π 3 e − i 5 8 · 4 π = ( p − v ) e − i σ W γ 1 ( a , p ) x − ℓ ( γ 1 ) = 0 3 3 c Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We write c ( γ ) for the contribution of the walk γ to the sum. One can partition the set of walks γ finishing at p , q or r into pairs and triplets of walks: γ 1 γ 2 γ 2 γ 1 γ 3 In the first case: c ( γ 1 ) + c ( γ 2 ) = ( q − v ) e − i σ W γ 1 ( a , q ) x − ℓ ( γ 1 ) + ( r − v ) e − i σ W γ 2 ( a , r ) x − ℓ ( γ 2 ) c c � � e i 2 π 3 e − i 5 8 · − 4 π + e − i 2 π 3 e − i 5 8 · 4 π = ( p − v ) e − i σ W γ 1 ( a , p ) x − ℓ ( γ 1 ) = 0 3 3 c Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We write c ( γ ) for the contribution of the walk γ to the sum. One can partition the set of walks γ finishing at p , q or r into pairs and triplets of walks: γ 1 γ 2 γ 2 γ 1 γ 3 In the first case: c ( γ 1 ) + c ( γ 2 ) = ( q − v ) e − i σ W γ 1 ( a , q ) x − ℓ ( γ 1 ) + ( r − v ) e − i σ W γ 2 ( a , r ) x − ℓ ( γ 2 ) c c � � e i 2 π 3 e − i 5 8 · − 4 π + e − i 2 π 3 e − i 5 8 · 4 π = ( p − v ) e − i σ W γ 1 ( a , p ) x − ℓ ( γ 1 ) = 0 3 3 c In the second case: c ( γ 1 )+ c ( γ 2 ) + c ( γ 3 ) � � 1 + x c e i 2 π 3 e − i 5 3 + x c e − i 2 π 3 e − i 5 = ( p − v ) e − i σ W γ 1 ( a , p ) x − ℓ ( γ 1 ) 8 · − π 8 · π = 0 . 3 c Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We deduce that the integral on any discrete contour vanishes. The next step is to choose the right contour. Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We deduce that the integral on any discrete contour vanishes. The next step is to choose the right contour. ε S T,L L cells β 0 a α T cells ¯ ε Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We deduce that the integral on any discrete contour vanishes. The next step is to choose the right contour. ε S T,L L cells β 0 a α T cells ¯ ε Sum the relation over all vertices in V ( S T , L ) (contour=boundary): � � F ( z ) + e i 2 π 3 � F ( z ) + e − i 2 π 3 � 0 = − F ( z ) + F ( z ) z ∈ α z ∈ ε z ∈ ¯ z ∈ β ε Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We deduce that the integral on any discrete contour vanishes. The next step is to choose the right contour. ε Definition: S T,L A x � x − ℓ ( γ ) , T , L = γ ⊂ S T , L : a → α \{ a } L cells � B x x − ℓ ( γ ) , T , L = β 0 a γ ⊂ S T , L : a → β α T cells � x − ℓ ( γ ) . E x T , L = γ ⊂ S T , L : a → ε ∪ ¯ ε ¯ ε Sum the relation over all vertices in V ( S T , L ) (contour=boundary): � � F ( z ) + e i 2 π 3 � F ( z ) + e − i 2 π 3 � 0 = − F ( z ) + F ( z ) z ∈ α z ∈ ε z ∈ ¯ z ∈ β ε Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We know the winding on the boundary! Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We know the winding on the boundary! For instance, the winding on the boundary β equals 0, so: � F ( z ) = B x T , L . z ∈ β Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice

We know the winding on the boundary! For instance, the winding on the boundary β equals 0, so: � F ( z ) = B x T , L . z ∈ β The same reasoning gives: F ( z ) = e i π F ( z ) = e − i π 8 8 � � 2 E x E x T , L T , L 2 z ∈ ε z ∈ ¯ ε � 3 π � � � A x F ( z ) = F ( a ) + F ( z ) = 1 − cos T , L 8 z ∈ α z ∈ α \{ a } Plugging these equalities into the previous relation, we obtain the claim. Hugo Duminil-Copin, Universit´ e de Gen` eve Self-avoiding walk on the hexagonal lattice