FLUCTUATIONS OF THE LOCAL MAGNETIC FIELD IN FRUSTRATED MEAN-FIELD ISING MODELS

Transcription

1 FLUCTUATIOS OF THE LOCAL MAGETIC FIELD I FRUSTRATED MEA-FIELD ISIG MODELS T. C. DORLAS AD W. M. B. DUKES Abstract. We consider fluctuations of the local magnetic field in frustrated mean-field Ising models. Frustration can come about due to randomness of the interaction as in the Sherrington-Kirkpatrick model, or through fixed interaction parameters but with varying signs. We consider central limit theorems for the fluctuation of the local magnetic field values w.r.t. the a priori spin distribution for both types of models. We show that, in the case of the Sherrington-Kirkpatrick model there is a central limit theorem for the local magnetic field, a.s. with respect to the randomness. On the other hand, we show that, in the case of non-random frustrated models, there is no central limit theorem for the distribution of the values of the local field, but that the probability distribution of this distribution does converge. We compute the moments of this probability distribution on the space of measures and show in particular that it is not Gaussian.. Frustrated Ising models and the local field distribution The celebrated Sherrington-Kirkpatrick model of a spin glass is given by the Hamiltonian H SK = i,j= J i,j s i s j,. where the s i = ± are Ising spins and the interaction parameters J i,j are i.i.d. random variables with Gaussian distribution. It was proposed and solved in [, ] by Sherrington and Kirkpatrick using the replica trick. However, their solution is flawed because it predicts negative entropy at low temperatures. An alternative solution scheme was proposed by Parisi [3, 4], which is generally regarded as being correct. However, it also involves the mathematically dubious replica trick and the mathematical status of the solution is therefore still unclear. Indeed, this model presents a considerable challenge to mathematicians [9]. evertheless, some progress has been made. Aizenman, Lebowitz and Ruelle [6] proved that, in the absence of an external field, the Sherrington-Kirkpatrick SK solution is correct in the high temperature domain. Pastur and Schcherbina [7] proved that the SK solution is correct unless the Edwards-Anderson order parameter is not self-averaging which implies the latter. Guerra [8] derived a beautiful inequality which implies that the the SK solution is correct in the high-temperature domain, 000 Mathematics Subject Classification. Primary: 8B44, Secondary 60B0, 60B05. Key words and phrases. Spin glasses, frustrated spin systems, probability measures on infinite-dimensional spaces, limit theorems, occupation measures.

2 T. C. DORLAS AD W. M. B. DUKES even in the presence of an external field, and recently, Talagrand [0, ] extended Guerra s bounds in various ways. A different class of models with frustrated interactions was introduced by Eisele and Ellis [4, 5, 6]. Their models have Hamiltonians of the form H EE = J i j s is j,. i,j= where Jx is a bounded continuous function. If this function takes on both positive and negative values, for example Jx = cosx, then the model is frustrated. A detailed study of these models was made in [4, 5] using large deviation theory. Here we consider the local magnetic field fluctuations for both types of models. Unsurprisingly, we find that the two models behave quite differently, but remarkably, the SK model behaves in a more regular way in that a central limit theorem holds with probability. More precisely, we define the occupation measure for the fluctuations by µ = i= δ j= J i,js j.3 and show that µ converges a.s. in the parameters J i,j, in probability with respect to the a priori flat distribution of the spins to a normal distribution as. This weak result which does not seem to have been observed before is a prerequisite for a large deviation property for these variables, which would be of interest in considering the thermodynamic limit of the model. The local fields play a pivotal role in mean-field models, see also the approach of Thouless, Anderson and Palmer [] and Mézard, Parisi and Virasoro [3]. On the other hand, in the models introduced by Eisele and Ellis we show that µ does not converge, even in probability. In fact, we consider a slight variant of their models, which does not quite fit within their class, but is more closely related to the SK model. It is given by the interaction parameters J i,j =, if 0 < i j M or i j M; 0, if i = j;, if M < i j < M = 3M..4 Here = 4M. If the measure.3 were to converge to a stable law γ then we would have E [e ] [ i f,µ δ γ e i f,µ ] = e i f,γ.5 for continuous functions f. We will show that the distribution of the measures µ does converge but.5 does not hold. Indeed, we compute all moments lim E[ f, µ k ] and show that the series converges for bounded continuous functions f. We then prove that this implies the convergence of the probability distribution of the measures µ to a nontrivial measure on

3 the set M R of probability measures on R. otice that for fixed ferromagnetic couplings J i,j = J, we have E e i µ,f = e i f J j= s j = e ifjx γdx.6 {s i } which is also not the same as.5. In this case the limiting distribution is a Gaussian distribution of δ-measures. Probability measures on the space of states were introduced by Aizenman and Wehr [7] for general translation-invariant random spin systems and their importance in the case of short-range spin-glasses has been argued at length by ewman and Stein [8, 9, 0]. Our example shows that such measures can be remarkably complicated even in very simple situations.. The local field distribution in the SK model We prove that µ γ in probability w.r.t. the spin configurations a.s. with respect to the randomness. This can also be formulated as follows: Theorem.. The random measures µ defined by.3 where the parameters J i,j are i.i.d. random variables with standard normal distribution satisfy E[F µ ] F γ a.s. with respect to the distribution of the coupling parameters, for any continuous function F on the space of probability measures M R with the topology of weak convergence, where γ is the Gaussian measure on R with mean zero and variance, i.e. γdx = e x / dx π. Proof. We first prove convergence for functions F of the form F µ = e i µ,f. This can be done by proving convergence of the moments E[ µ, f p ]. To prove that these converge to γ, f p almost surely, we compute E[ µ, f p ] γ, f p,.7 where the overline denotes the average over the random couplings. In case of long expressions, we also use the notation [ ]. First consider the first moment p =. We put fx = e itx so that γ, f = e t / = ct, and compute ow, E[ µ, f ] γ, f = j= t cos J i,j = and similarly j= j= i= j= t cos J i,j ct e i t J i,j e i t J i,j = j= 3..8 e t = ct.9 t t cos J i,j cos J i,j = ct.0

4 4 T. C. DORLAS AD W. M. B. DUKES if i i and otherwise j= t cos J i,j = = j= j= e i t J i,j e i t J i,j 4 [ ] e it J i,j = e t.. Hence E[ µ, f ] γ, f = i,i = j= t t cos J i,j cos J i,j ct [ = ] e t ct. t t4 e t e t e t4 / = O..3 By the Borel-Cantelli lemma we can now conclude that, almost surely, E e itx µ dx e itx γdx.4 and by interchanging the order of integration and the fact that convergence of characteristic functions implies weak convergence of bounded measures see [], E[ µ, f ] γ, f a.s..5 for all bounded continuous functions f. The same strategy applies also to higher moments. In that case we consider [ p ] E µ, f α α= p γ, f α.6 α=

5 5 with f α x = e itαx. As above we get [ p ] E µ, f α α= = p = p p γ, f α α= i,...,i p= j= i,...,i p i,...,i p cos [ p j= p t α J iα,j α= p ct α α= cos t J i,j t p J ip,j cos t J i,j t p J i p,j p ct α α= ] p ct α..7 We distinguish three cases. The first case is when all i α and all i β are different, i.e. #{i,..., i p, i,..., i p} = p. These contribute nothing to the above sum because of.9. The second case is when one pair is equal, i.e. #{i,..., i p, i,..., i p} = p. In this case there are two possibilities: either #{i,..., i p } = #{i,..., i p} = p and there exists one pair α, β such that i α = i β or {i,..., i p } {i,..., i p} = but #{i,..., i p } = p and #{i,..., i p} = p or vice versa. The second possibility again gives no contribution because the two factors in the last expression of.7 separate and we can use.9 again in one of the factors. In case #{i,..., i p } {i,..., i p} = we can assume i α = i β and we have j= = cos t J i,j t p J ip,j cos t J i,j t p J i p,j exp i t α t β J iα,j j= exp i p t α t β J iα,j α =,α α p α =,α α j= t α J iα,j t α J iα,j p β =,β β p β =,β β t β J i t β J i β,j β,j = e tαt β e tα t β e t t p t α t β /..8 If #{i,..., i p, i,..., i p} p then we simply bound the two factors in the right-hand side of.7 by = 4. The total number of terms in the sum in this case is bounded by [ 3 p 4 ] p p,.9 3

6 6 T. C. DORLAS AD W. M. B. DUKES and the number of terms in the second case is less than p p. We thus obtain [ p ] p E µ, f α γ, f α α= 4 [ 3 α= ] p p 4 3 p e tαt β / e tα t β / α,β= e t α t β / = O..0 As in the case of p = above, we conclude that E[ fx,..., x p µ dx... µ dx p ] fx,..., x p γdx... γdx p a.s. for all bounded continuous functions f. In particular,. E[ µ, f p ] γ, f p. a.s. for all bounded continuous functions f. By expanding the exponential, it then follows that E[e i µ,f ] e i γ,f..3 Unlike the finite-dimensional situation, this is not sufficient for the convergence of the measures. By Prokhorov s theorem [], we need to prove tightness of the sequence of probability distributions of the µ. This is done in the following lemma. Tightness of the sequence of probability measures on M R follows from: Lemma.. For all ɛ > 0 there exists a compact set K ɛ M R such that for all, Pµ / K ɛ < ɛ.4 a.s. with respect to the coupling parameters. Proof. We define K n = { µ M R µ[, a a, ] n 4 e a a }..5 Clearly, K n is compact. By Chebyshev s inequality, P µ [, a a, ] > n 4 e a < n 4 e a E µ,, a a,..6 We now bound, a a, by e x a e xa and compute It follows that E µ, e ±x = i= E e ± j= J i,js j = e..7 P µ [ a, ] > n 4 e a n 4 e a..8 By the Borel-Cantelli lemma, we conclude that, for n large enough, P µ [, a a, ] > n 4 e a < n e a.9

7 7 a.s. for all a. The lemma now follows from Pµ K c n for n large enough. P µ [, a a, ] > n 4 e a a= e n a= e a = e e n < ɛ The local field distribution in the frustrated model 3.. The first and second moment. For the first moment, the convergence is easy: Let us introduce the notation L {s x } = f, µ = f x= y= J x,y s y. 3. Then L {s x } = {s x} M k= M 4M k f. 3. M k To see this, write y= J x,ys y = x y=x M s y xm y=x s y x3m y=xm s y and sum over the possible values k of this variable with possible number of occurrences at fixed x. The sum over s x gives an additional factor. Hence lim E [ µ, f ] = π R / fxe x dx = f, γ. 3.3 Therefore, if.5 were to hold the limiting measure γ would be a standard normal distribution. Computation of the higher moments is more complicated. We first consider the case k =. For the second moment, we need to compute the limit lim L {s x }. 3.4 {s x} ow L {s x } = f J x,ys y f J x,ys y. 3.5 x,x {s x} {s x} y y

8 8 T. C. DORLAS AD W. M. B. DUKES To compute the limit of this expression we consider it as a quadratic form and insert e it z and e it z for f. We then need to compute lim = lim x,x {s x} [ ] i exp t J x,y t J x,ys y x,x y s y=± y e i / t J x,yt J x,ys y 3.6 = lim cos t J x,y t J x,y x,x y ow, if x x M, the number of y for which J x,y = J x,y is M x x and the number of y for which J x,y = J x,y is x x. Except of course if x = x, but this case is negligible as we are dividing by. Similarly, the cases y = x, x are irrelevant. On the other hand, if x x > M then the number of y with J x,y = J x,y is x x M and the number of y with J x,y = J x,y is x x. Thus, the above limit equals. lim k=m { M k= = lim M k k cos / t t cos / t t k M } k cos / t t cos / t t k=m { M = { exp 0 exp k=0 k t t k t t This can be rewritten as k t t t t [ ] { st t st t } ds k } [ ] } {st t st t } ds. 3.7 e t t / The limit 3.4 is therefore given by e st t du. 3.8 dx dx ρ x, x fx fx, 3.9

9 where ρ is the density of the measure with characteristic function given by 3.8, i.e. 9 dx dx ρ x, x e it x t x = e t t / e st t ds. 3.0 Diagonalising t st t t we find ρ x, x = 4π e x x x x 4 s 4s ds. 3. s Clearly, this is not equal to the density of γ γ, i.e. π e x x / which would be the result if.5 were to hold Figure. Plot of the function 3.. It appears from this plot that ρ is constant on squares, i.e. it only depends on x x. This will be proved in the appendix.

10 0 T. C. DORLAS AD W. M. B. DUKES 3.. Higher moments. The computation of the higher moments follows the same strategy but the result is more complicated and cannot be expressed in such a simple fashion. Analogous to 3.5 we have L {s x } p = {s x} x,...,x p {s x} i= Inserting exponentials e it iz we have to compute lim p = lim p x,...,x p {s x} x,...,x p p f J xi,ys y. 3. [ i exp cos y y y s y i= ] p t i J xi,y p t i J xi,y. 3.3 We now show that a calculation as in the case of the second moment yields the following: Theorem 3.. For bounded continuous functions f, the limit exists and equals lim E [ f, µ p ] = lim i= p L {s x } p {s x} dx... dx p ρ p x,..., x p fx... fx p, where the probability density ρ p is given by ρ p x,..., x p = π p/ 0 dα... dα p... 0 det Sα,..., α p e x, Sα,...,α p x 3.4 and the matrix Sα,..., α p has matrix elements sα i α j, i, j =,..., p, where 4 α if α <, sα = α 3 if α.. Proof. First notice that for all pairs i < j, #{y : J xi,y = J xj,y} = { xj x i if x j x i M, x j x i if x j x i > M. 3.6

11 .B. The right-hand side is correct up to an error of at most, which is irrelevant in the limit. We now rewrite the limit 3.3 as follows: p lim p t i J xi,y x,...,x p y i= = lim p exp p t i t j J xi,yj xj,y x,...,x p y i,j= = e t t p lim p exp t i t j J xi,yj xj,y x,...,x p i<j y = e t t p lim p exp t i t j sx j x i / x,...,x p i<j The last expression follows from the fact that J xi,yj xj,y = #{y : J xi,yj xj,y = } #{y : J xi,yj xj,y = } y = Taking the limit now yields e t t p 3.7 { xj x i x j x i if x i x j M, x j x i x j x i if x j x i > M. 0 dα dα p exp [ t i t j sα i α j ]. 3.9 The result 3.4 then follows from the well-known Fourier transform formula for Gaussian functions. The formula 3.4 can be simplified by a transformation of variables. We subdivide the domain of integration into subdomains as follows. First let u j := α j α and define s j := su j for j =,..., p. Lemma 3.. Let π be a permutation of {,..., p } and let σ,..., σ p {±}. Define the region Rπ, σ [0, ] p by u,..., u p Rπ, σ iff i<j 0 u π 4 σ π < < u πp 4 σ πp. 3.0 Then the region Rπ, σ is equivalent to < σ π s π < < σ πp s πp <. 3. and the elements of the matrix S are given by S ii = for all i p S i = S i = s i for all < i p S ij = S ji = σ j s i σ i s j σ i σ j if σ i s i < σ j s j.

12 T. C. DORLAS AD W. M. B. DUKES Moreover, if we denote b i := σ πi s πi, then p det Sα,..., α p = p b b p b i b i. 3. Proof. The equivalence of regions follows immediately from 3.5 which implies { 4uk, if σ s k = k = 4u k 3, if σ k = which can be written as i= s k = 4σ k u k σ k 3.3 from which u k 4 σ k = 4 σ ks k. 3.4 To determine the matrix elements of S, notice first that S i,i = s0 = and S i, = S,i = s u i = s i for i >. Moreover, if < i < j, S i,j = S j,i = s u j u i, so we may assume σ i < σ j s j. ow, by 3.3 we have su i u j = 4σ u i u j σ, 3.5 where σ = if u i u j and σ = otherwise. By the above equivalence, 0 u i 4 σ i < u j 4 σ j and hence 4 σ j σ i u j u i 4 σ j σ i. 3.6 From this it is easy to see that { uj u u i u j = i, if σ i σ j u i u j, if σ i > σ j and This implies σ = {, if σi σ j,, if σ i < σ j. and σ = σ i σ j 3.7 Inserting these identities into 3.5 we obtain u i u j u i u j = σ i σ j σ. 3.8 su i u j = 4σ i σ j u i u j σ i σ j σ j σ i, which is the stated result. To evaluate det S we perform several elementary row and column operations and show the resulting matrix to be the matrix B, given in the

13 appendix. For all i p multiply each entry in row i by σ i and each entry in column i by σ i. otice that if < i < j p and σ i s i < σ j s j, then the resulting matrix S satisfies S ij = σ i σ j σ j s i σ i s j σ i σ j = σ i s i σ j s j = b π i b π j. The entries in the row now read b π... b π p. Reorder the rows and columns, according to the permutation π, so that the b indices are increasing in the first row and column. The resulting matrix is B. The total number of row and column operations is even, due to the symmetry of the matrix S, so the sign of the determinant is preserved. Thus det S = det B, which is evaluated in the Lemma A. in the appendix. The inverse of the matrix B can also be worked out: see Lemma A. in the Appendix. This leads to the following representation of the density ρ p : Corollary 3.3. The density ρ p of 3.4 can be written as where ρ p x 0, x,..., x p = p π p/ g p x 0, x,..., x p ; σ, π, 3.9 gx 0, x,..., x p ; σ, π = σ,π v v p 4 v i 0, i 3 π/ dv... dv p dα 3.30 π/ { exp x0 σ π x π v x 0 σ πp x πp 4 p 3.3 i= v i sin α p σπi x πi σ πi x πi 3.3 i= v i } σπp x πp σ πp x πp 4 p vi sin α In particular, g 3 x 0, x, x ; σ, π is given by g 3 x 0, x, x ; σ, π = 0 π/ dv 0 dα π/ { [ exp x0 σ π x π v0 x 0 σ π x π v0 / sin α ]} σ πx π σ π x π v0 / sin α Figure 3. shows a contour plot of the density ρ 3 at fixed x from which it is apparent that the simple property of ρ mentioned in the caption of Figure 3. does not generalise to higher p.

14 4 T. C. DORLAS AD W. M. B. DUKES Figure. Contour plot of the function ρ 3 x 0, x, 0.6 generated using the integrals in Convergence of the probability distribution on the space of measures. The convergence of the characteristic functions E [ e i f,µ ] follows immediately from the theorem of Section 3, but as in Section Lemma., this does not yet imply that the corresponding distributions converge. This therefore requires a proof: Theorem 3.4. The sequence of probability distributions of the measures.5 converges weakly to the probability distribution on M R with characteristic function given by E [ e i f, ] = p=0 i p p! This theorem follows from dx... dx p ρ p x,..., x p fx... fx p Lemma 3.5. For all ɛ > 0, there is a compact K ɛ M R and 0 such that Pµ K ɛ < ɛ for all = 4M with M. Proof. Let K ɛ := { µ M R } µ[, a a, ] < e a /4 a. ɛ Clearly this K ɛ is compact, since for all δ > 0 there exists a > such that µ[ a, a] c δ for all µ K ɛ, i.e. K ɛ is tight. Using Chebychev s inequality

15 5 we have P µ [, a a, ] > π e a /8 ɛ < ɛ E µ,, a a, e a / where, as in 3. with fx =, a a, E µ, a, = L {s x } = = {s x} 4M 4M M k= M M k=[a M] 4M M k 4M. M k k, a a, 4M We now use the bounds 4M M < 4M M for all M and M i Mi for all i M, which follows from e x > x, to bound the coefficients 4M Mk < 4M M exp k 4M E µ, infty, a a, <. This gives the following bound = 4M M k=[a M] M a/ < e a /4. Applying this to 3.36 gives the result: P µ a, π e a /8 ɛ This yields as before Pµ K c ɛ a M e u / du < exp i 4M 4M k exp M 4M e x /4M dx < ɛ e a /8 π. P µ [, a a, ] > π e a /8 ɛ a= ɛ π a= e a /8 ɛ e u /8 du = ɛ π 0 Proof. of Theorem 3.4. By Prokhorov s theorem see [] the lemma implies that the set of probability measures {µ } is relatively compact. This means that every subsequence has a convergent subsequence which must

16 6 T. C. DORLAS AD W. M. B. DUKES have the characteristic function given by 3.35 and is therefore uniquely determined. It follows by the usual subsequence argument that the sequence µ itself must converge to this measure. Appendix Lemma A.. The value ρ x, y given in 3. depends on x y only. Proof. otice that it is clearly symmetric under interchange of x and y and also under sign change of x or y. We can therefore assume that 0 x < y. Differentiating w.r.t. x then yields d dx ρ x, y = 4π otice that the exponent can be rewritten as x sxyy x sy e s s ds 3/ x sy s y. Since x < y, x sy has a zero inside the integration interval [, ]. We therefore divide this range into the intervals [, x/y] and x/y, ]. On the second interval we change variables to s in such a way that whereas and Solving the latter yields s = = s = and s = x/y = s = x/y x sy s = x s y s ds s = ds s s = c s s c s s and inserting the boundary conditions then gives y x c =. y x A simple calculation shows that the other identity also holds. The integral over the interval x/y, ] now transforms into minus the integral over [, x/y so that the two contributions cancel and the derivative is zero. Lemma A.. Let B be the symmetric matrix with entries: B ii := for i p B,i = B i, := b i for i p B i,j = B j,i := b i b j, for i < j p. Then det B = p b b b... b p b p b p. Define b 0 :=, b p := and w i := b i b i for all i = 0,..., p. Then the

17 7 inverse B is given by: B, = w p w 0 Bi,i = w i w i for i p B, = B, = w 0 B i,i = B B,p = B i,i = w i for i p p, = w p. Proof. The determinant is obtained by elementary row and column operations: subtract row from all other rows; then add column to column ; finally successively subtract column i from column i for i =,..., p. The resulting matrix is upper triangular and the product of diagonal elements is the said value. To prove the statement about the inverse of B, we multiply row x of B and column y of B and consider various cases. If x = y = we have BB = b p w p w 0 w p b w 0 =. If x = y = we have BB = b w 0 w 0 w b b w =, and if x = y >, BB yy = b y b y w y w y w y b y b y w y =. The,- and, elements are: BB = w 0 b w 0 w b w = 0 and BB = b b p w p b w 0 w p w 0 = 0. For y > we get BB y = b y w y b y w y w y b y w y = 0 and BB y = b y b p w p b y w p w 0 b b y w 0 = 0. For the cases x y, first assume, < x < y < p. Then BB xy = b x b y w y b x b y w y w y b x b y w y = 0. If y = p we have BB xp = b x b p w p b x b p w p w p b x w p = 0. If < y < x < p, BB xy = b y b x w y b y b x w y w y b y b x w y = 0. Finally, if < y < p, BB py = b y b p w y b y b p w y w y b y b p w y = 0 and BB p = b p w 0 b b p w 0 w b b p w = 0. Lemma A.3. The density ρ p x,..., x p may be expressed as ρ p x 0, x,..., x p = p π p/ σ,...,σ p ± π P erm[,p ] gx 0, x,..., x p ; σ, π

18 8 T. C. DORLAS AD W. M. B. DUKES where the sum is over all permutations π of {,..., p } and π/ gx 0, x,..., x p ; σ, π = dv... dv p dα v v p 4 v i 0, i π/ { exp x0 σ π x π v x 0 σ πp x πp 4 p i= v i sin α p σπi x πi σ πi x πi i= v i } σπp x πp σ πp x πp 4 p. vi sin α Proof. Because of periodicity of the function sα the p-fold integral in 3.4 can be transformed into the following p -fold integral over u i = α i α. Since s is an even function of u, we have that p regions are similar and we obtain ρ p x,..., x p = p π p/ du... du p [0,] p det Su,..., u p { exp } x, S u x ext, we perform a change of variables to s. By 3.3 we have ds i /du i = 4σ i, and the above integral may be written as ρ p x,..., x p = p ds... ds p π p/ σ,π σ π s π σ πp s πp det S s, σ, π { exp } x, S s, σ, π x. ow change variables to b. This yields ρ p x,..., x p = p π p/ σ,π db... db p det S b, σ, π { exp } x, S b, σ, π x b b p We now recall that det S b, σ, π = p b b b... b p b p b p. Moreover, the scalar product x, S b, σ, π x simplifies by reordering the vector x according to the permutation π, multiplying each entry by it s corresponding σ value and then using the matrix B rather than S. Thus we have x 0, x,..., x p, S b, σ, π x 0, x,..., x p = x 0, σ π x π,..., σ πp x σp, B x 0, σ π x π,..., σ πp x σp A brief calculation shows that this value is indeed x 0 σ π x π b x 0 σ πp x πp b p p i= σ πi x πi σ πi x πi. b i b i

19 9 and so gx 0, x,..., x p ; σ, π = exp b b p db... db p x 0 σ π x π / b x 0 σ πp x πp / b p p i= σ πix πi σ πi x πi /b i b i p b b b... b p b p b p. Finally, we perform the following change of variables. For shorthand in this proof, let V := v...v p. For i p, let b i = v v i / and b p = 4 V 4 V sin α. Then the Jacobian is b,...,b p = v,...,v p,α 4 v v... v p 4 V cos α. This is also the value of the denominator in the integral when expressed in the new variables. Thus the two cancel to leave only an exponential term. Under this change of variables, the region { b R p b b p } is equivalent to the region { v R p, α v, π/ α π/}. The resulting integral is just the one stated in the lemma, which is also References [] D. Sherrington & S. Kirkpatrick, Phys. Rev. Lett. 35, [] S. Kirkpatrick & D. Sherrington, Phys. Rev. B 7, [3] G. Parisi, Phys. Rev. Lett. 43, [4] G. Parisi, J. Phys. A. 3, [5] G. Parisi, J. Phys. A 3, [6] M. Aizenman, J. Lebowitz & D. Ruelle, Commun. Math. Phys., [7] L. Pastur & M. Shcherbina, J. Stat. Phys. 6, [8] F. Guerra, Comm. Math. Phys. 33, 003. [9] M. Talagrand, Prob. Th. & Rel. Fields 0, [0] M. Talagrand, Ann. Prob. 8, [] M. Talagrand, Ann. Prob. 30, [] D. J. Thouless, P. W. Anderson & R. G. Palmer, Phil Mag. 35, [3] M. Mézard, G. Parisi & M. A. Virasoro, Europhys. Lett., [4] Th. Eisele & R. S. Ellis, J. Phys. A 6, [5] Th. Eisele & R. S. Ellis, Z. Wahrsch. Geb [6] Eisele, Th. Equilibrium and nonequilibrium theory of a geometric long-range spin glass. Critical Phenomena, Random Systems and Gauge Theories. K. Osterwalder & R. Stora eds., Les Houches, 984, 55 65, orth-holland, Amsterdam, 986. [7] M. Aizenman & J. Wehr, Commun. Math. Phys. bf 30, [8] C. M. ewman & D. L. Stein, Phys. Rev. Lett. 76, [9] C. M. ewman & D. L. Stein, Phys. Rev. E 55, [0] C. M. ewman & D. L. Stein, J. Stat. Phys [] P. Billingsley, Convergence of Probability Measures. Wiley, ew York 995. Thm 6.3. []. Bourbaki, Eléments de Mathématique. Fasc. XXXV. Integration, Chap. IX. Hermann, Paris Dublin Institute for Advanced Studies, School of Theoretical Physics, 0 Burlington Road, Dublin 4, Ireland.