Loops, trees and operators Yves Le Jan Universit Paris-Sud May - PowerPoint PPT Presentation

Loops, trees and operators Yves Le Jan Université Paris-Sud May 2010

ENERGY and GREEN FUNCTION Graph, with conductances and killing measure: � � e ( f, f ) = 1 C x,y ( f ( x ) − f ( y )) 2 + κ x f ( x ) 2 2 x,y x � � λ x f ( x ) 2 − = C x,y f ( x ) f ( y ) x x,y with λ x = κ x + � y C x,y and C x,x = 0 . Under a transience assumption, the associated Green function is defined: G ( x, y ) = G ( y, x ) = [( M λ − C ) − 1 ] x,y with M λ := multiplication by λ . Also, given a ”discrete one form”: ω x,y = − ω y,x � � C x,y e iω x,y f ( x ) f ( y ) λ x f ( x ) f ( x ) − e ( ω ) ( f, f ) = x x,y Green function: G ( ω ) ( x, y ) = G ( ω ) ( y, x ) = [( M λ − Ce iω ) − 1 ] x,y

LOOP MEASURE and loop functionals y = C x,y Energy e → λ -symmetric transition matrix P x λ x → Markov chain → Bridge measure P x,y . t 1 with mass p t ( x, y ) = p t ( y, x ) = λ y [exp t ( P − I )] x,y . There is also � ∞ µ x,y = P x,y dt t 0 with mass G ( x, y ) and the σ − finite loop measure � ∞ � 1 t P x,x µ ( dl ) = ( dl ) λ x dt t 0 x ∈ X Occupation field of l � T ( l ) l x = 1 � 1 { l ( s )= x } ds λ x 0

LOOP MEASURE and loop functionals Number of jumps from x to y in l : N x,y ( l ) N x,y ( l ) µ ( dl ) = C x,y µ x,y ( dl ) and � l x µ ( dl ) = µ x,x ( dl ) In particular, � � � l x µ ( dl ) = G ( x, x ) N x,y ( l ) µ ( dl ) = C x,y G ( x, y ) and LOOP ENSEMBLE: L (or L 1 ) = Poisson Point Process with intensity µ � � L x = � � l x N x,y ( L ) = N x,y ( l ) l ∈L l ∈L n n � � � � L x i ) = E ( G ( x i , x σ ( i ) ) 1 σ ∈S n 1

RANDOM SPANNING TREE Finite graph with κ � = 0 : Add a ”cemetery point” δ . Spanning trees are rooted in δ . Generalized Cayley Theorem: � � C x,y = det( G ) spanning trees τ ( x,y ) edge of τ with the convention C x,δ = κ x . → P ST probability on spanning trees. Sampling by Wilson algorithm, based on loop erasure. Erased loops = Loop ensemble L

FREE FIELD The complex free field φ ( x ) is defined as the complex Gaussian field with covariance G ( x, y ) E ( φ ( x 1 ) ...φ ( x m ) φ ( y 1 ) ...φ ( y n )) = δ nm Per ( G ( x i , y j )) Bosonic Fock space structure: L 2 ( σ ( φ )) is isomorphic to the Hilbert space F B generated by a "vacuum" vector 1 and creation/anihilation operators a x , a ∗ x , b x , b ∗ x with [ a x , a ∗ y ] = [ b x , b ∗ y ] = G ( x, y ) and all others commutators vanishing. Then φ ( x ) and φ ( x ) are represented by two dual commuting operators a x + b ∗ x and a ∗ x + b x : � � for any polynomial P, 1 , P ( φ, φ )1 F B = E ( P ( φ, φ )) Anihilation operators can be interpreted in terms of functional ∂ ∂ derivatives: a x = ∂φ ( x ) , b x = ∂φ ( x )

GRASSMANN FIELD Anticommuting variables ψ x , ψ x are defined as operators on the Fermionic Fock space F F generated by a vector 1 and creation/anihilation operators c x , c ∗ x , d x , d ∗ x with y ] + = [ d x , d ∗ y ] + = G ( x, y ) and with all others [ c x , c ∗ anticommutators vanishing. Then ψ x = d x + c ∗ ψ x = − c x + d ∗ x x Note that ψ x is not the dual of ψ x , but there is an involution I on F F such that ψ = I ψ ∗ I � � 1 , ψ ( x 1 ) ...ψ ( x m ) ψ ( y 1 ) ...ψ ( y n )1 = δ nm det( G ( x i , y j )) On a finite graph, ψ x , ψ x can also be defined in terms of differential forms. ”Supersymmetry” between φ and ψ : for any polynomial F � � 1 , F ( φφ − ψψ )1 F B ⊗F F = F (0) ( 1 denotes 1 ( B ) ⊗ 1 ( F ) )

ISOMORPHISMS (Wilson Algorithm) Loop ensemble L ← → Random Spanning Tree � � Free field φ, φ ← → Grassmann field ψ, ψ (”Supersymmetry”)

The Bosonic isomorphism Given two energy forms e, e ′ with C ′ ≤ C , λ ′ ≥ λ and a ”discrete one form” ω x,y = − ω y,x : det( G ′ [ C ′ ( ω ) ) � e iω x,y ] N x,y ( L 1 ) e − P ( λ ′ x − λ x ) b x,y L x ) = E ( C x,y det( G ) x,y � � 1 , exp( e ( φ, φ ) − e ′ = ( ω ) ( φ, φ ))1 F B In particular for C = C ′ and ω = 0 , we get that � � � E ( e − P ( κ ′ x − κ x ) b L x ) = ′ 1 , exp( ( κ x − κ x ) φ x φ x ))1 Therefore the fields φφ and � L have the same joint distributions.

Variational Identities Given n distinct vertices x i ∂ n ∂ n E ( F (1 E ( F ( � � ∂κ x i L )) = � ∂κ x i 2 φφ )) � L x i − G ( x i , x i )]) = E ( F ( � [ � L ) i � [ | φ ( x i ) | 2 − G ( x i , x i )]) = E ( F ( φφ ) i � � � � E ( F ( � l i ) − F ( � µ x i ,x i ( dl ) L + L )) = In fact, more generally � E ( F ( L ∪ l i ) − F ( L )) � µ x i ,x i ( dl ) ∂ n Q ∂κ xi E ( F ( L )) =

Variational Identities L y − N x,y ( L ) − N y,x ( L ) L x + � If we set T x,y ( L ) = � = G x,u + G y,v − G x,v − G y,u , K ( x,y ) , ( u,v ) then E ( T x,y ( L )) = E ( | φ ( x ) − φ ( y ) | 2 ) = K ( x,y ) , ( x,y ) Given n DISTINCT edges ( x i , y i ) , we get a 2nd variational formula: ∂ n ∂ n E ( F (1 E ( F ( � � ∂C x i ,y i L )) = � ∂C x i ,y i 2 φφ )) � = E ( F ( � [ T x i ,y i ( L ) − K ( x i ,y i ) , ( x i ,y i ) ]) L ) i � = E ( F (1 [ | φ ( x i ) − φ ( y i ) | 2 − K ( x i ,y i ) , ( x i ,y i ) ]) 2 φφ ) i � � � � E ( F ( � l i ) − F ( � [ µ x i ,x i ( dl ) − 2 µ x i ,y i ( dl )+ µ y i ,y i ( dl )] = L + L ))

The fermionic isomorphism � C ′ � κ ′ det( G ) x,y e iω x,y x E ST ( ) = det( G ′ C x,y κ x ( ω ) ) ( x,y ) ∈ τ x, ( x,δ ) ∈ τ � � 1 , exp( e ( ψ, ψ ) − e ′ = ( ω ) ( ψ, ψ ))1 F F The Transfer Current Theorem follows directly � P ST (( x i , y i ) ∈ τ ) = det( K ( x i ,y i ) , ( x j ,y j ) ) C x i ,y i � In particular, P ST (( x i , δ ) ∈ τ ) = det( G ( x i , x j )) κ x i . NB: φ and ψ can also be used jointly to represent bridge functionals: In particular � � � F ( � l ) µ x,y ( dl ) = 1 , φ x φ y F ( φφ − ψψ )1 F B ⊗F F � � = 1 , ψ x ψ y F ( φφ − ψψ )1 F B ⊗F F

IN THE CONTINUUM Domain D ⊂ R d , or Riemannian manifold with metric and killing rate � � e ( f, f ) = 1 a i,j ( x ) ∂f ∂f k ( x ) f ( x ) 2 det( a ) − 1 det( a ) − 1 2 dx + 2 dx 2 ∂x i ∂x j ( − 1 2∆ x + k ( x )) G ( x, y ) = δ y ( x ) with ∆ x = � a i,j ( x ) ∂ 2 f ∂x i ∂x j Bridge measures and σ − finite measure µ on Brownian loops are defined in the same way (Lawler and Werner ”loop soup”).

IN THE CONTINUUM Occupation field � l : for d = 1 , local times. For d ≥ 2 random measure, defined on test functions. For d = 1 the fields φ and ψ are defined in the same way as on graphs. For d ≥ 2 they are generalized field i.e. defined only on test functions). L x − G ( x, x )” is well defined on test functions For d = 2 and 3 , ” � x , l i ∈ L . as the compensated sum of the � l i Associated by a version of the Bosonic isomorphism with the Wick square of the free field ” | φ ( x ) | 2 − G ( x, x )” (square for the tensor product structure). L x can be defined and For d = 2 , n -th renormalized powers of � are associated with Wick 2 n -th powers of φ .

IN THE CONTINUUM Problems of definitions in higher dimensions, or for the field T , even in d = 1 . → They should be interpreted as unbounded operators, on adequate domains: e.g., in any dimension, for any point x inside a domain D , | φ ( x ) | 2 − G ( x, x ) operates on polynomials in the Fock space associated to the boundary ∂D . Some variational identities are defined and valid for adequate functionals of L and for variations of the killing rate k or of the inverse metric a i,j

IN THE CONTINUUM For d = 1 , Loop ensemble L = ensemble of Brownian excursions (from the minima, or the maxima of the loops) Describes the history of a (quadratic) continuous branching process with immigration. For d = 1 , the determinantal process = determinantal processes formed by the points x i with independant spacings such that ( x i , δ ) ∈ τ exists defined by Macchi. For constant killing rate, G ( x, y ) = ρ exp( − | x − y | /α ) . Interpretation of the law of the spacings: � 1 − 2 ρα x K e − x α sinh( α ) . For d = 2 . SLE (2) trees linking any finite set of sites and SLE(8) contour (LSW).

Addendum: OTHER LOOP FUNCTIONALS l x and A loop l in L includes more information than the fields � N x,y ( l ) .

Hitting distributions i and G D denotes the Green If F 1 and F 2 are disjoint, if D i = F c function of the chain killed outside D The probability that no loop in L intersects F 1 and F 2 equals, l ( F 2 ) > 0 } )) = det( G D 1 ) det( G D 2 ) exp( − µ ( { � l ( F 1 ) � (1) det( G ) det( G D 1 ∩ D 2 ) >From that, by some matrix manipulations ∞ � 1 2 kTr ([ H 12 H 21 ] k + [ H 12 H 21 ] k ) µ ( � l ( F 1 ) � l ( F 2 ) > 0) = (2) 1 with H 12 = H F 2 | F 1 (hitting distributions of F 2 from F 1 ) and H 21 = H F 1 | F 2 . The k -th term of the expansion can be interpreted as the measure of loops with exactly k -crossings between F 1 and F 2 .