RC Expansion/Edwards Sokal Measure

Transcription

1 RC Expansion/Edwards Sokal Measure April 11, 2007 Let Λ d be finite and consider the q state Potts model with couplings {J x, } and free boundar condition (no boundar condition). The Hamiltonian (q = 2 is Ising) is H() = J x, (δ x, 1), where x {1,..., q}. The partition function is [] = e βh() = e β[jx,(δx, 1)] (1) Now consider the random cluster model with parameter q and bond weights {p x, } with partition function given b [] = q c() (p x, ) (1 p x, ), (2) : x,=1 : x,=0 where c() denotes the number of connected components of the bond configuration. Note that if we set e βjx,(δx, 1) = δ x, + (1 δ x, )e βjx, = δ x, p x, + (1 p x, ), p x, = 1 e βjx,. Plugging this into (1), we see that e βh() = (δ x, p x, + (1 p x, )). (3) 1

2 If we expand out the product, we get (δ x, p x, ) Ω Ω(Λ) Ω / Ω (1 p x, ), where Ω(Λ) denotes the set of bonds connecting points in Λ. Each term above can be identified as a bond configuration where each bond is occupied if and onl if it lies in Ω. The above sum can now be rewritten as (δ x, p x, ) (1 p x, ). : x,=1 The partition function (1) now becomes (δ x, p x, ) : x,=1 : x,=0 : x,=0 (1 p x, ). (4) Observe that the above sum is finite and so far is simpl summed over all possible bond and spin configurations, and hence we ma interchange the order of summation to obtain (δ x, p x, ) (1 p x, ). (5) : x,=1 : x,=0 From (4) we see that if is such that a bond is present when x, then the contribution to the sum is zero. Similarl, in (5), we see that if is such that x when there is a bond present between x and, then the contribution to the sum is zero. So to get rid of the δ x,, we introduce the constraint function (, ) = () = (), which is the indicator function of the fact that and are compatible in the sense that x = whenever x, = 1 and x, = 0 whenever x. We can now rewrite the Potts partition function as [] = () (p x, ) (1 p x, ). : x,=1 : x,=0 Now we observe that given, () = q c(), (6) 2

3 since the functions forces each connected cluster of to have the same spin and there are q possible spins. We can now perform the inner sum in the penultimate displa to see that in fact (with p x, = 1 e βjx, ) [] = [] [,]. So this quantit is the normalization constant for three (probabilit) measures: Potts, random cluster, and Edwards Sokal, and from now on will simpl be denoted. Explicitl, the Potts measure assigns a spin configuration the probabilit µ P () = 1 (δ x, p x, + (1 p x, )), the random cluster measure assigns a bond configurations the probabilit µ RC () = 1 (δ x,,1p x, + δ x,,0(1 p x, )) and the Edwards Sokal measure assigns a spin bond configuration (, ) the probabilit µ ES (, ) = 1 (, ) (δ x,,1p x, + δ x,,0(1 p x, )). (7) B (6), if we fix and sum (7) over all spins, then we get exactl µ RC () (so the marginal distribution of the bond variables is the RC model). Similarl, if we fix and sum (7) over all bonds (putting in δ x, and erasing (, )), then we get 1 (1 p x,) + p x, δ x, ), which b (3) is exactl µ P () (so the marginal distribution of the spin variables is the Potts model). Now let s find the conditional distributions. Let g() be a (dumm) spin observable. We compute the expected value of g(), g() Λ f, with free boundar conditions in Λ. We would like to write it as a nested expectation E [E (g() )]. Writing out g() λ f and expanding as before using (3), 3

4 we have g() Λ f = 1 e βh() g() = 1 [δ x, p x, + (1 p x, ] g() = 1 () [δ x,,1 p x, + δ x,,0 (1 p x, )] g() = 1 [δ x,,1 p x, + δ x,,0 (1 p x, )] (, )g() = 1 q c() [δ x,,1 p x, + δ x,,0 (1 p x, )] (, )g() (, ) = ( ) (, ) µ RC (), (, )g() where we have used (6) ( (, ) = qc() ). So we see that µ ES ( ) = 1 (, ), qc() that is, given, the conditional measure concentrates on s which are compatible with, assigning each such probabilit 1/q c(). Equivalentl, given, the clusters of are labeled 1,..., q with uniform probabilit. Here is a quick application of this: consider q = 2 (the Ising case). Then x Λ f = ( ) (, ) µ RC () 2 c() (2δ x, 1). If x and and in the same connected component of, then x = and the contribution of the inner sum is 1. If x and are in different connected components, then with probabilit 1/2 (so half the contributing configurations) the will be labeled the same spin, in which case the inner sum contributes -1/2. Therefore the net contribution from bond configurations with x and in different clusters is 0, and we conclude x Λ f = µ RC(x ). 4

5 This formula can be generalized to integer values of q, if we take the tetrahedron representation, i.e. we represent the spin variables as unit vectors pointing to vertices of a (q 1) dimensional tetrahedron, so that we have x = (qδ x, 1)/(q 1). Now let g() be a (dumm) bond observable. Then we ma reverse expand using (3) and (6) and get g() = 1 q c() [δ x,,1 p x, + δ x,,0 (1 p x, )] g() = 1 (, ) [δ x, δ x,,1 p x, + δ x,,0 (1 p x, )] g() = 1 (, ) δ x, p x, (1 p x, ) g() : x,=1 : x,=0 = 1 x, [p x,δ x, + (1 p x, )] (, ) x, [δ x,,1 p x, + δ x,,0 (1 p x, )] x, [δ x,,1 p x, + δ x,,0 (1 p x, )] (, )g() = e βh() ( (, ) x, [δ ) x,,1 p x, + δ x,,0 (1 p x, )] (, ) x, [δ x,,1 p x, + δ x,,0 (1 p x, )] g(). So we see that µ ES ( ) = (, ) x, [δ x,,1 p x, + δ x,,0 (1 p x, )] (, ) x, [δ x,,1 p x, + δ x,,0 (1 p x, )]. Again the conditional measure is concentrated on bond configurations which are compatible with : Given, bonds are placed between x and with probabilit p x, if the have the same spin. References [1] M. Aizenman, J. T. Chaes, L. Chaes, and C. M. Newman. Discontinuit of the Magnetization in One-Dimensional 1/ x 2 Ising and Potts Models. J. Stat. Phs. 77, (1994). 5