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Potts and O n non lin model in StatMech OSP Spanning Forest correspondence

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80 pages
Potts and O(n) non-lin. ?-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” Universite Paris XIII January 19th 2010 S. Caracciolo, A.D. Sokal and ?A. Sportiello -? Clifford representation of the Forest Algebra

  • spanning-forest correspondence

  • cluster model

  • forest algebra

  • temperley-lieb

  • linear ?-models

  • ?i ?

  • variables ?i


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Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
representation of an algebra ated to spanning forests
Clifford rel
Andrea Sportiello work in collaboration with S. Caracciolo and A.D. Sokal
Seminar at “Laboratoire d’Informatique de Paris-Nord” Universit´eParisXIII January 19th 2010
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Potts andO(n) non-linearσ-model in Statistical Mechanics Potts andO(n) non-linearσ-models More on Potts: the Random Cluster Model More onO(n and): supersymmetryOSP(n|2m) Models
OSP(1|2) – Spanning-Forest correspondence The theorem Thermodynamic properties Robustness ofOSP(1|2) symmetry for interacting forests
A Clifford representation of Temperley-Lieb Linear-space dimension of the polynomial algebra GettingRabcd= 0 fromRbca= 0, fromRab= 0 Even/odd Temperley-Lieb and Partition Algebras
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Potts and O(n) non-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models
Potts andO(n) non-linearσdom-els
IPotts Model:variablesσi∈ {0,1, . . . ,q1}; exp(βH(σ)) = expPhhijiJijδ(σi, σj)i Symmetry: ‘global’ permutations inSq. IO(n) non-linearσ-model:variablesiRn; exp(βH(σ)) =Qi2δ(|σi2| −1)expPhhijiwij(1ij)i Symmetry: ‘global’ rotations inO(n)(continuous!). IIf21(ij)21instead of (ij1): extra ‘local’Z2symmetryiiσi, with=±1. ~ 1 In other words, the’s are in theprojective space:RPn. hRPn1:=x~Rnr{0}~xλx~i
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Potts and O(n) non-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models
Potts andO(n) non-linearσsledom-
IPotts Model:variablesσi∈ {0,1, . . . ,q1}; exp(βH(σ)) = expPhhijiJijδ(σi, σj)i Symmetry: ‘global’ permutations inSq. IO(n) non-linearσ-model:variables~iRn; σ exp(βH(σ)) =Qi2δ(|σi2| −1)expPhhijiwij(1ij)i Symmetry: ‘global’ rotations inO(n)(continuous!). IIf12(ij)21instead of (ij1): extra ‘local’Z2symmetry~~σwith±1 iiσi,= . In other words, the’s are in theprojective space:RPn1. hRPn1:=~xRnr{0}x~λ~xi
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Potts and O(n) non-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models
Potts andO(n) non-linearσlesm-do
IPotts Model:variablesσi∈ {0,1, . . . ,q1}; exp(βH(σ)) = exphPhijiJijδ(σi, σj)i Symmetry: ‘global’ permutations inSq. IO(n) non-linearσ-model:variables~iRn; σ exp(βH(σ)) =Qi2δ(|σi2| −1)expPhhijiwij(1ij)i Symmetry: ‘global’ rotations inO(n)(continuous!). IIf12(ij)21instead of (ij1): extra ‘localZ2symmetryiii, with=±1. In other words, the’s are in theprojective space:RPn1. hRPn1:=~xRnr{0}~xλ~xi
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Some goals:
I
I
Potts and O(n) non-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models
Find relations between Potts andO(n) non-lin.σ-models, and with combinatorial “generating functions” (i.e. countings of graphical structures);
Understandanalytic continuationinqfor Potts Model, and innforO(n);
IUnderstandcomputational complexityfor the generating function (and existence ofFPRAS), as a fn. ofqand ofn;
I
Understandasymptotic freedomin a geometric and non-perturbative way, inD= 2 Euclidean lattice, for our ‘favourit ’ odel: Potts [q0;J/qfixed] e m O(n) non-linσ-model [n→ −1]Spanning Forests.
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Some goals:
I
I
I
I
Potts and O(n) non-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models
Find relations between Potts andO(n) non-lin.σ-models, and with combinatorial “generating functions (i.e. countings of graphical structures);
Understandanalytic continuationinqfor Potts Model, and innforO(n);
Understandcomputational complexityfor the generating function (and existence ofFPRAS), as a fn. ofqand ofn;
Understandasymptotic freedomin a geometric and non-perturbative way, inD= 2 Euclidean lattice, for our ‘favourite’ model: Potts [q0;J/qfixed] O(n) non-linσ-model [n→ −1]Spanning Forests.
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra
Potts andO(n)non-lin.σ-model in StatMech OSP(1|2) – Spanning-Forest correspondence A Clifford representation of Temperley-Lieb
Some goals:
I
I
I
I
Potts and O(n) non-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models
Find relations between Potts andO(n) non-lin.σ-models, and with combinatorial “generating functions” (i.e. countings of graphical structures);
Understandanalytic continuationinqfor Potts Model, and innforO(n);
Understandcomputational complexityfor the generating function (and existence ofFPRAS), as a fn. ofqand ofn;
Understandasymptotic freedomin a geometric and non-perturbative way, inD= 2 Euclidean lattice, for our ‘favourite’ model: Potts [q0;J/qfixed] O(n) non-linσ-model [n→ −1]Spanning Forests.
S. Caracciolo, A.D. Sokal andhA. Sportielloi
Clifford representation of the Forest Algebra