Clifford representation of an algebra related to spanning forests - PowerPoint PPT Presentation

Potts and O ( n ) non-lin. σ -model in StatMech OSP(1 | 2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb Clifford representation of an algebra related to spanning forests Andrea Sportiello work in collaboration with S. Caracciolo and A.D. Sokal Seminar at “Laboratoire d’Informatique de Paris-Nord” Universit´ e Paris XIII January 19th 2010 S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech OSP(1 | 2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb Potts and O ( n ) non-linear σ -model in Statistical Mechanics Potts and O ( n ) non-linear σ -models More on Potts: the Random Cluster Model More on O ( n ): supersymmetry and OSP ( n | 2 m ) Models OSP (1 | 2) – Spanning-Forest correspondence The theorem Thermodynamic properties Robustness of OSP (1 | 2) symmetry for interacting forests A Clifford representation of Temperley-Lieb Linear-space dimension of the polynomial algebra ac = 0, from R ab = 0 Getting R abcd = 0 from R b Even/odd Temperley-Lieb and Partition Algebras S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Potts and O ( n ) non-linear σ -models ◮ Potts Model: variables σ i ∈ { 0 , 1 , . . . , q − 1 } ; � � � exp( − β H ( σ )) = exp � ij � J ij δ ( σ i , σ j ) Symmetry: ‘global’ permutations in S q . σ i ∈ R n ; ◮ O ( n ) non-linear σ -model: variables � � � � � � exp( − β H ( σ )) = � 2 δ ( | σ 2 i | − 1) exp � ij � w ij (1 − � σ i · � σ j ) i Symmetry: ‘global’ rotations in O ( n ) (continuous!). � � σ j ) 2 − 1 ◮ If 1 ( � σ i · � instead of ( � σ i · � σ j − 1): 2 extra ‘local’ Z 2 symmetry � σ i → ǫ i � σ i , with ǫ = ± 1. σ ’s are in the projective space: RP n − 1 . In other words, the � � � � � � x ∈ R n � { 0 } RP n − 1 := � � x ∼ λ� x S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Potts and O ( n ) non-linear σ -models ◮ Potts Model: variables σ i ∈ { 0 , 1 , . . . , q − 1 } ; � � � exp( − β H ( σ )) = exp � ij � J ij δ ( σ i , σ j ) Symmetry: ‘global’ permutations in S q . σ i ∈ R n ; ◮ O ( n ) non-linear σ -model: variables � � � � � � exp( − β H ( σ )) = � 2 δ ( | σ 2 i | − 1) exp � ij � w ij (1 − � σ i · � σ j ) i Symmetry: ‘global’ rotations in O ( n ) (continuous!). � � σ j ) 2 − 1 ◮ If 1 ( � σ i · � instead of ( � σ i · � σ j − 1): 2 extra ‘local’ Z 2 symmetry � σ i → ǫ i � σ i , with ǫ = ± 1. σ ’s are in the projective space: RP n − 1 . In other words, the � � � � � � x ∈ R n � { 0 } RP n − 1 := � � x ∼ λ� x S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Potts and O ( n ) non-linear σ -models ◮ Potts Model: variables σ i ∈ { 0 , 1 , . . . , q − 1 } ; � � � exp( − β H ( σ )) = exp � ij � J ij δ ( σ i , σ j ) Symmetry: ‘global’ permutations in S q . σ i ∈ R n ; ◮ O ( n ) non-linear σ -model: variables � � � � � � exp( − β H ( σ )) = � 2 δ ( | σ 2 i | − 1) exp � ij � w ij (1 − � σ i · � σ j ) i Symmetry: ‘global’ rotations in O ( n ) (continuous!). � � σ j ) 2 − 1 ◮ If 1 ( � σ i · � instead of ( � σ i · � σ j − 1): 2 extra ‘local’ Z 2 symmetry � σ i → ǫ i � σ i , with ǫ = ± 1. σ ’s are in the projective space: RP n − 1 . In other words, the � � � � � � x ∈ R n � { 0 } RP n − 1 := � � x ∼ λ� x S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Some goals: ◮ Find relations between Potts and O ( n ) non-lin. σ -models, and with combinatorial “generating functions” (i.e. countings of graphical structures); ◮ Understand analytic continuation in q for Potts Model, and in n for O ( n ); ◮ Understand computational complexity for the generating function (and existence of FPRAS), as a fn. of q and of n ; ◮ Understand asymptotic freedom in a geometric and non-perturbative way, in D = 2 Euclidean lattice, for our ‘favourite’ model: Potts [ q → 0; J / q fixed] ≡ O ( n ) non-lin σ -model [ n → − 1] ≡ Spanning Forests. S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Some goals: ◮ Find relations between Potts and O ( n ) non-lin. σ -models, and with combinatorial “generating functions” (i.e. countings of graphical structures); ◮ Understand analytic continuation in q for Potts Model, and in n for O ( n ); ◮ Understand computational complexity for the generating function (and existence of FPRAS), as a fn. of q and of n ; ◮ Understand asymptotic freedom in a geometric and non-perturbative way, in D = 2 Euclidean lattice, for our ‘favourite’ model: Potts [ q → 0; J / q fixed] ≡ O ( n ) non-lin σ -model [ n → − 1] ≡ Spanning Forests. S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Some goals: ◮ Find relations between Potts and O ( n ) non-lin. σ -models, and with combinatorial “generating functions” (i.e. countings of graphical structures); ◮ Understand analytic continuation in q for Potts Model, and in n for O ( n ); ◮ Understand computational complexity for the generating function (and existence of FPRAS), as a fn. of q and of n ; ◮ Understand asymptotic freedom in a geometric and non-perturbative way, in D = 2 Euclidean lattice, for our ‘favourite’ model: Potts [ q → 0; J / q fixed] ≡ O ( n ) non-lin σ -model [ n → − 1] ≡ Spanning Forests. S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Some goals: ◮ Find relations between Potts and O ( n ) non-lin. σ -models, and with combinatorial “generating functions” (i.e. countings of graphical structures); ◮ Understand analytic continuation in q for Potts Model, and in n for O ( n ); ◮ Understand computational complexity for the generating function (and existence of FPRAS), as a fn. of q and of n ; ◮ Understand asymptotic freedom in a geometric and non-perturbative way, in D = 2 Euclidean lattice, for our ‘favourite’ model: Potts [ q → 0; J / q fixed] ≡ O ( n ) non-lin σ -model [ n → − 1] ≡ Spanning Forests. S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra

Potts and O ( n ) non-lin. σ -model in StatMech Potts and O(n) non-linear σ -models: an intro OSP(1 | 2) – Spanning-Forest correspondence More on Potts: the Random Cluster Model A Clifford representation of Temperley-Lieb More on O(n): OSP(n | 2m) Models Analytic continuation is easy for Potts... [Fortuin-Kasteleyn (1972), relating Potts p.fn. to the Tutte Poly.] � � � � � � � e − β H ( σ ) = v ij := e J ij − 1 Z G = 1 + v ij δ ( σ i , σ j ) σ σ ( ij ) � � � � � � = v ij δ ( σ i , σ j ) σ H ⊆ G ( ij ) ∈ E ( H ) ( ij ) ∈ E ( H ) � � comp. �� � � q K ( H ) = v ij . K ( H ) = # in H H ⊆ G ( ij ) ∈ E ( H ) Recognize the (slightly reparametrized and rescaled) multivariate Tutte Polynomial of G , and even better on next slide... S. Caracciolo, A.D. Sokal and � A. Sportiello ✍ � Clifford representation of the Forest Algebra