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

Andrea Sportiello

Description

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

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 14
• Langue: English

Extrait

Potts and O (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. Sportiello✍i

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-linearσ-model in Statistical Mechanics Potts and O (n) non-linearσ-models More on Potts: the Random Cluster Model More on O (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 GettingR abcd = 0 fromR b ca = 0, fromR ab = 0 Even/odd Temperley-Lieb and Partition Algebras

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

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-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models

Potts and O (n) non-linearσdom-els

I Potts Model:variablesσ i ∈ {0,1, . . . ,q−1}; exp(−βH(σ)) = exp P h hiji J ij δ(σ i , σ j ) i Symmetry: ‘global’ permutations inS q . I O (n) non-linearσ-model:variables~σ i ∈R n ; exp(−βH(σ)) = Q i  2δ(|σ i 2 | −1)  exp P h hiji w ij (1−~σ i ∙~σ j ) i Symmetry: ‘global’ rotations in O (n)(continuous!). I If 2 1  (~σ i ∙~σ j ) 2 −1  instead of (~σ i ∙~σ j −1): extra ‘local’Z 2 symmetry~σ i → i σ i , with=±1. ~ −1 In other words, the~σ’s are in theprojective space:RP n . h RP n−1 :=  x ~ ∈R n r{0}   ~ x∼λx ~ i

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

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-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models

Potts and O (n) non-linearσsledom-

I Potts Model:variablesσ i ∈ {0,1, . . . ,q−1}; exp(−βH(σ)) = exp P h hiji J ij δ(σ i , σ j ) i Symmetry: ‘global’ permutations inS q . I O (n) non-linearσ-model:variables~ i ∈R n ; σ exp(−βH(σ)) = Q i  2δ(|σ i 2 | −1)  exp P h hiji w ij (1−~σ i ∙~σ j ) i Symmetry: ‘global’ rotations in O (n)(continuous!). I If 1 2  (~σ i ∙~σ j ) 2 −1  instead of (~σ i ∙~σ j −1): extra ‘local’Z 2 symmetry~~σwith±1 i → i σ i ,= . In other words, the~σ’s are in theprojective space:RP n−1 . h RP n−1 :=  ~ x∈R n r{0}  x ~ ∼ λ~ x i

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

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-linearσ-models: an intro More on Potts: the Random Cluster Model More on O(n): OSP(n|2m) Models

Potts and O (n) non-linearσlesm-do

I Potts Model:variablesσ i ∈ {0,1, . . . ,q−1}; exp(−βH(σ)) = exp h P hiji J ij δ(σ i , σ j ) i Symmetry: ‘global’ permutations inS q . I O (n) non-linearσ-model:variables~ i ∈R n ; σ exp(−βH(σ)) = Q i  2δ(|σ i 2 | −1)  exp P h hiji w ij (1−~σ i ∙~σ j ) i Symmetry: ‘global’ rotations in O (n)(continuous!). I If 1 2  (~σ i ∙~σ j ) 2 −1  instead of (~σ i ∙~σ j −1): extra ‘localZ 2 symmetry~σ i → i ~σ i , with=±1. ’ In other words, the~σ’s are in theprojective space:RP n−1 . h RP n−1 :=  ~ x∈R n r{0}   ~ x∼ λ~ x i

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

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

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 innfor O (n);

I Understandcomputational 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 [q→0;J/qfixed] e m ≡ O (n) non-linσ-model [n→ −1]≡Spanning Forests.

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

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

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 innfor O (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 [q→0;J/qfixed] ≡ O (n) non-linσ-model [n→ −1]≡Spanning Forests.

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

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

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 innfor O (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 [q→0;J/qfixed] ≡ O (n) non-linσ-model [n→ −1]≡Spanning Forests.

S. Caracciolo, A.D. Sokal andhA. Sportiello✍i

Clifford representation of the Forest Algebra