Manifolds in random media: a variational approach to the spatial probability distribution

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Manifolds in random media: a variational approach to the spatial probability distribution

Yadin Goldschmidt

To cite this version:

Yadin Goldschmidt. Manifolds in random media: a variational approach to the spatial probability distribution. Journal de Physique I, EDP Sciences, 1994, 4 (11), pp.1583-1596. �10.1051/jp1:1994209�.

�jpa-00247016�

J. Phys. IFrance 4 (1994) 1583-1596 S O V E M B E R 1994. PAGE 15E3

Classification P h y s m Abstracts 05.40 ^~ 05.20 ^~ 75.10N

Manifolds in random media: a variational spproach to the spatial probability distribution

Yadin Y. Goldschmidt

Department of Physics and Astronomy, University of Pittsburgh: Pittsburgh, PA 15260, U.S.A.

(Received 2,5 April 1991, received in final form 9 August 1994, accepted 12 August 1994)

Abstract. - We dcvciop a new variaiional scheme to approximate tlie p o d i o n dependent spatial probability distribution of a zero dimensional manifold in a random mcdium. This celebrated 't,oy-model' is associated via a. ma,pping with directed polymers in 1+1 dimension, and also describes feat.ures of the commensiirate-incommensurate phase transition. I t consists of a pointlike 'intrrfare' in one dimension subject to a coinbiiiation of a harmonic potential plus a random potential with long range spatial correlations. The variational approach we develop gives far better results for the tail of the spatial distribution than the Harniltonian version, dcvrloped by Mezard and Pirisi, is compared with numerical siniulations for a range of tempcraturcs.

This is because the variational parameters arc deteiniined a functions of position. The replica method i s utilized, and solut,ions for the rariationd parmeters are presented. In this paper we limit, ourselves t o the r q l i c a synimetiic solution.

1. Introduction.

Recently a lot of attention has been devoted to t h e behavior of manifolds in random media [I-211. This is partially due t o tlicir connection with vortex line pinning in high 7, supercon- ductors [ll, 211: ^{b u t} also because of the intrinsic interest in t,he behavior of interfaces between two coexisting phases of a disordered system. like magnets subject t o random fields or random impiirit,ies. In addit,ion mappings are known to exist between one dimensional manifolds like directed polymers in disordered media and growth problems in t h e presence of random noise described by t,he K P Z equation [22].

The variational method [8, 13, 1Sj appears to b e an important tool in approximating many propert,ies of t h e sysaem like the calculatiori of t h e roughness exponent of a wandering manifold i n a disordered medium. Recently we have conduct,ed a numerical st,udy of t h e spatial proba- bility distribution of directed polymers in I f 1 dimensions im t h e prcsence of quenched disorder [15]. Directed polymers refer t o an interface in t,wo dimensions wit,h no overhangs. In a,ddition to t h e width of t h e interface, it is important t o know what is t h e probability t h a t the interface would wander a cert,ain distance in the transverse direction. We have then exploited a mapping

1584 JOURNAL DE PHYSIQUE I NO11

of this system into that of a pointlike interface in one dimension subjected to a combination of a harmonic restoring force and a random potential with long ranged correlations.

Using that mapping combined with a use of the variational approximation, we were able to derive many properties of the spatial probability distribution (PD) of the directed polymers [16]. In particular the function f(a) associated with the decay of the probability for different realizations of the disorder has been derived using the replica method. Replica symmetry breaking solutions have been used (at low temperaturesj. The mapping from directed polytiiers to the so called toy-model is discussed in detail in reference 1161 based on previous work by Parisi 151 and Bouchaud and Orland 16;. In particular the spatial probability distribution of directed polymers at large times relates t o the spatial probability distribution of the toy-model at low temperature. Villain e t al. [I] emphasize directly the relation between the toy-model and the cominensurate-inconimensurate transition in systems with quenched impurities [l]

Using the replica inethod the toy-model can be mapped into a n n-body problem were the interaction beta-een different particles (replicas) is given by the correlation of the original random potential. The variational method utilizes ail effective quadratic Namiltonian whose parameters are chosen by the stationarity method to give a good approximation for the free energy of the n-body system in the h i t 71 i 0 [8]. The results it provides for the tail of the spatial distribution are not very accurate as will be shown below (Sect. 4). In this paper we describe a way to directly apply the variational approach t,o the probability distribution rather than to the Hamiltonian or the free energy. As we shall see it gives a niuch better approximation for the behavior of the tail of the distribution. For high temperatures the results appear almost exact without the need to break replica symmetry.

2 . The variational method applied to the probability distribution.

The toy model [I, 4, 13_ 161 involves a classical particle in a oiie-dimensional potential consisting of a harmonic part and a spatially correlated random part. It is the simple prototype of an interface ~- a point in one dimension ^~ which is localized by a harmonic restoring force, arid pinned by quenched impurities the distribution of which may be correlated over large separations. Its Hamiltoniaii is given by:

P z (1)

N ( w ) = -w 2 - t V ( w j ;

whcre w denotes the position of the particle and V ( w ) is a random potential with a Gaussian distribution characterized by:

{ v ( w ) ) = 0: (t/(w)tT(w')) = --9 / w - ~ ' / * ' ~ 7 + ^{Const. - j((u -U')'), (2)}

2t1 - Y)

where the brackets indicate averaging over the random potential. In the sequel without loss of generality we will talie t,hc constant on the r.h.s. of equation (2) to be zero. We will make use of the replica method to replicate the partition function

2 = /d. exp( - f i H ) ₍₃₎

N O 1 1 MANIFOLDS IN RANDOM MEDIA 1585

and average it over the random potential to express it in terms of an effective n-body Haniil- tonian:

( Z " ) = /dwl . - . d u n exp(-P7-1) ,

For later use let us d s o define an n-body quadratic Hamiltonian:

where the matrix elements of crab are free parameters at this point.

We consider cases in which the potentid has long-range correlations, (i.e. y < I), and in particular we are interested in the case 7 = 4 because of the above mentioned mapping from directed polymers in 1 + ¹ dimensions. Let us review briefly the variational scheme used by Mezard and Parisi. We start with the well known inequality

(eA) 2 e (a) (6)

The Mezard-Parisi variational approximation [S] can be obtained from this inequality, by choos- ing

(7)

- - A ( q , . 1 , . ,U-) = FL(w1,. ..,U,,) - h ( q , ' . . ,wn)

fl

And by defining the average as

Tr A exp ( -ph ) (A' = Trexp (-oh) ^' with

+m

Tr =

1,

This yields

(10)

1 1

nTrexp (-87-1) 5 (7-1-h) - - In Trexp ( - P h ) , nf = - - I P

P

where f is the free energy. The variational free energy is given by the right hand side of equatioii (10) at the point of stationarity. This procedure provides equations for the variational parameters which are the elements of the matrix uab. MP 1131 have solved these equations and fonnd a replica symmetric (RS) solution valid for high temperatures and a solution which breaks replica symmetry (RSB) a t low temperatures.

Recall that the limit. n t 0 has to be taken when making use of the replica method. An important point to bear in mind is that the ineqnality (6) holds only when n is a positive integer, provided the parametrization of u is such that Tr exp(-ph) < 00. This is because only in this case the integration measure

[ Trexp(--ph)]-'exp (-bh) ^dwl . . . dw,: ₍₁₁₎ is a real positive inewure on R", for which the proof of the inequality (6) hoids (see e.g. [23] p.

61). In the limit n ⁱ 0 the inequa1it.y can change sign and does not hold in general, but one

1586 JOURK.4L DE PHYSIQUE I NO1 I

still expects the stationary point of the r.1i.s. of equation (10) to give a good approximation to the 1.h.s. i.e. to t,he exact free energy. Support for this contention comes also from the fact that t,he variational method becomes exact for a manifold enlhedded in d spat.ia1 dimensions wheii d becomes infinite. and one can systeniatically improve it by a l / d expansion [8: 13; I T , 191.

Let iis dcfine

5 ⁼ / d u i . - - d w , 6(wi - w) (12)

A

The function

Ph ^{( U )} = Tr exp ( - p h j , ₍₁₃₎

(with the parameters u m ~ determined by the stationarity conditions which were discussed above) constitutes an approximation to the exact spatial probability distribution. averaged over the random realizations of the potential, arid given by the formula

exp(-PH(w)) = ;i;; exp (-px ^{) ~ (14)}

d o exp( -BII(c))

i

PfI ( d ) =

(i

with the limit, n

-

with

Gab = [ j d - i? 1;;. (16)

Substituting t,he RS and RSB solutions for Gi1 olitained in reference [13], we fiiid [16]

with the 'reduced' temperature f defined as

f < 1 (17)

In the sequel we refer to the expressions giveri in equation (17) as "the Hamiltonian variational approximation''

I proceed to derive a new variational scheme which is more appropriate for approximating the position-dependent spatial probability dist.ribution. The idea is to use the inequality ( 6 ) , but substitute 6 from equation (12) for the trace iii equation (8). This means; defining

P11 MANIFOLDS IN RANDOM MEDIA 1587

The notation (0) is used to distinguish the w-dependent average from the averaged defined iii equation (8). Usiiig A = - p ( X - h ) I find

I

PH(ij ⁵ Trexp(-BX) 2 ^exp ( - / 3 ( ( X - h ) ) ( w ) ) x s e x p ( - p h ) Pt,(w) ( 2 0 )

I

Again, for positive integers n, and Trexp ( -@h) < the measure

[ S e x p ( - p h ) ] - ’ e x p (-,Bh) 6(wl - w) dwl . . . dw n , (21) is a positive measure on R” for any value of w and the inequality (6) holds. For 7~ + 0 this is no longer true, but the stationary point of t,he r.h.s. of equation (ZO), denoted by Pv(w):

with respect to the parameters uab is expec.ted t o yield a good approximation to the spatial probability distribution by virtue of analytic continuation. This will be checked in section 4 in coniparison with numerical sirnulations. Such a variational determination of P,(w) is obtained for each value of w separately aiid thus it is expected to provide a better approximatioii than by just using the ua5 obtaiiied globally from extremizing the free energy (see Eq. (lo)),

and using equation (13). One should note that it does not follow from our discussion that P,(w) is properly normalized. Provided it is a good approximation to the exact probability distribution, its normalization will be close to one, and correcting for the exact normalization will have practically no effect on the behavior in t,he tail region were the probability is very small and where the variational calculation is most important.

Our next task is to evaluate the quantity ((X - h ^))(U) needed in order to calculate P v ( w ) , see equation (20). If we use the integral representation for the Dirac 6-functioii in equation (12):

(22)

/

6(wi - U ) = we can write

In the Appendix we sliour how the integrals c x i be evaluated for general n. The end result. is:

with

1588 JOURNAL DE PHYSIQUE I NO11

For the speanl case of interest y = 1 / 2 (see Eq (2) for j) uie find ( ( K - @ / ( U ) =

where erf is the usual error function. Equations (24) and (26) together with the expression (20) for P,(w) constitnte the main results of this section. In the rest of the paper we consider only the important case of y = 112.

3. The replica symmetric solution.

In this section we consider the replica symmetric solution to the variational stationarity equa- tions. In this case we search for a solution for which all the off diagonal elements of the rnairix oab are equal ~7 well as all the diagonal elements:

we also define

In terms of these parameters one finds

Tliils w e have two variational paranieters a and LA, as contrasted with t,he Hamiltonian vari- ational approach were there is only one variable because in that case translationid invariance dictates p1 = li. In addition, the variational parameters are of course dependent on cd in the present case. We define for convenience:

r = cqll + ^{G ~ ~ .} ( 3 0 ) Using the result of the previous section, equation (26); we can express the probability distri- bution P,,(w) (see also Eqs.(,lO) and (13)) as

where the limit TL i 0 has been taken. The stationary points of equation (31) in the U - @I

plane as functions of ^{U ,} have been obtained numerically for various values of the parameters

N " l 1 MANIFOLDS IN RANDOM MEDIA 1589

of the model ( , 8 > p , g ) , and the results have been used back in equation (31) to evaluate P u ( w ) . The results will be summarized below. But first let us examine the simpler behavior in the tail of the probability distribution, that can be investigated amalytically.

For large values of U, the error-functiori in equation (31) can be approximated by sign(w) and the exponential term in t,he expression for ^g(w) is negiigible. If we further define the tail region as the region for which

iUi >> ' 9 / P ⁱ (32)

we find that the extremuni of P,(w) is obtained for

Plugging these expressions back into the expression for P u ( w ) one finds t,hat the heliavior of the spatial probability distribution in the tail region defined by equation (32) is

where fil is given by equatioii ( 3 3 ) .

The extremuni in the tail turns out to be a minimum in the U - 1~~ plane, as opposed t,o the siauation for n 2 1 when it is a maximum as can he seen from equation ( 2 0 ) . (On the contrary, for very small values of w we have found numerically, that the stationarity point is actually a saddle point.) The fact that, the extremum in the tail region is a minimum seems to suggest that the approximate expression equation (34) constitntes an upper bound to the ex& behavior. The behavior in the tail girren by equation (34) sliould be compared with the behavior obtained from the conventional \-ariational approximation given by substituding

= 1 / 2 in equation (17). Those formulas give a higher value for the tail t,lian ihat predicted by equation (34). (This is true even when f' < 1 with the RSB solution). More on this in the discussion below.

4. Comparison with numerical siniulations.

We proceed to a numerical comparison betxeen siniulations and the results of the various variational approximations. Again, we limit the discussion to the case y = 112. We have studied numerically a lat,tice version of the boy model. ..1 suit,able interval of the particle's position w ( -12.5/fl i w < 1 2 . 5 / f l ) is divided into 5,000 lattice sites. For a giveiveu realization, the algorithm generates an independently distributed Gaussian random number for each site ^{T , :} it then generates V,, t,he correlated random potential at site j ; by summing the random numbers in the following may:

V, ⁱⁱ sign(i - j ) r , . ( 3 5 )

The quadratic term of the Hamiltonian is added to this random potential, and then the partition function and probability distribution are calculated.

We consider three sets of values for the parameters ( 9 , , 9 . p ) : (D.2,2&, l ) , (l .0,2& , i), (10.0,2.?.3.6). The corresponding values of the reduced temperature ? for these three cases

1590 JOURNAL DE PHYSIQUE I

0 200 400 600 800

0 2

Fig. 1. - Plot of t.he log of spatial probability distiibution vs. w z for f = 5 (solid line). The dashed line is the result of the IIamiltonian variational approximation and the diamonds the results of our new mriatianal scheme.

are 5 , 1 and 0.23 respectively. The data for f = 5 is plotted in figure 1 (solid curve). It

wils obtained from numerical simulations with 50000 realizations of the disorder. The dashed line represents the probability Ph given in equation (17) and derived from t,he Hamiltoniaii variational method. The diamonds represent the approximation developed in the last sectioii.

We see that it gives a perfect fit to the data for this value of f. We have used Matheiiiatica t o fiiid the s t a t i o n a q point of equation (31) in the pi - U plane aiid them substituted these values back into equation (31). The asymptotic formula equation (34) predicts

P,(d) =exp(-0.1u2 + 0 . 1 4 & - 2 . i ) (36)

wkiere we used p,i = 0.7 from equation (33). This behavior is also imdistiriguisliable from the data in the range 10 < w < 28."

Let us proceed to the case T = 1. The data is plotted i n figure 2. In this case the noise associxted with the random potential is apparent. In this case we found it necessary to average over 500000 realizations. The data for 50000 realizations is also shown in lighter dots. In the tail region it is apparent that the curve corresponding t o the lower number of realizations has a lower value, since the average is increased by relatively rare events. The dashed curve is the single paranleter Hamiltonian variational fit. The mew variational method results are represented by the diamonds. The asymptotic formulas predicts p I = 0.06 and

P,(w) 2 exp(-0.5u2 + ^{3.545, - 17).} (37)

It is represented by the solid line in the figure. The diamonds d o not lie exactly on this line, because w is still not large enough in t,his range. Again we see that the variational approximation gives ari excellent fit to the data.

The final example is for f = 0.23 using 106 realizations of the disorder. T h e d a t a is depicted in figure 3. The dashed line is t,he result of the Hamiltonian R.5 variation. T h e dot-dashed line

N " l 1 MANIFOLDS IK RANDOM MEDIA 1591

0 20 40 60 80 100 120 140 160 w 2

Fig. 2. - Plot of t h e log of the sparial probability distribution vs. w' for ? = 1. The wiggly solid

C U I V ~ represent. the result of averaging over 500000 realizat.ions. The light wiggly curve is for 50000 rcalizations. The dashrd curve and the diamonds are explained in the caption of figure 1. The solid smooth curve result from the asymptotic formula, equation (34).

is the result, of the RSB Hamiltonian variation which is the appropriate solution for < 1.

T h e diamonds again represcnt our new variat,iond method results. T h e range of w simulated is lower than t,he onset of the tail region given by eqiiation (3%) which in the present case starts abovc w

-

We see that, the d a t a in this case Pails below the result of our variational method, which is nonetheless bctter than the Harniltonian RSB variational approximation. Two possibilities come to mind t o explain this discrepancy.

1. It seems q u i k possible that. because the relat,ive strength of the randoni part of the poteiitial is niuch larger when T is small, as compared to the harmonic part,, one needs more rea1izat.ions t o acciiniiilate enough statistics for the avcragcd probability dist.ribution. For T = 5 even 5000 realizations already gave us good results. In figure I we show the results for 50000 rcalizations, but t,hese are practically indistinguishable from the ar-emge of 5000 realizations. For f' = 1, 50000 realizations a,re riot sufficicnt and me had to average at least 500000 realizations to accumulate eiioiigh statistics. This is because we have t o collect enough rare event,s which contribute to the distinction between the average and typical values of the spatial probability distribution [15. 161. It is quite plausible that to get bettcr results for

? = 0.23 one has to go t o a higher number of realizations. -4veraging over morc realizations (see e.g Fig. 2 for the distinction between 50000 and 500000 realizations) may narrow the gap between the d a t a and t,he variatioiid approximation.

2. Aiiother strong possibility is that like the Hamiltonian variational case it is necessary for small T t o look for a solution wit.h rcplica symmetry breaking when using the current va:ariational scheme. Such a solution may display a different, asymptotic behavior for small T than the one gireii hy equaiion (34). It may also +ld a lower value in the pretail region (which is t,he range of values depicted i n Fig. 3 ) . In order to find such a solutioii one has t,o go hack to

1592 JOURNAL DE PHYSIQUE I NO11

-200 ^I

0 2 4 6 8 10 12 14 16

0 2

Fig. 3. ^~ Plot of the log of the spatial probability distribution vs. w z for 'f = 0.23. The wiggly solid curve represent the result of averaging over 106 realizations. The dashed and dashed-dotted lines represent the Hamiltonian RS and RSB approxima,tions respectively. Thc diamonds are t h e results of the new variational scheme. The open triangles represent t h e empirical cubic approximation.

t h e full expressions derived in section 2 without making the simplifying assumptions of replica symmetry made in section 3, and one has to extremize t h e probability distributions under t h e more general conditions. The task of finding of a RSB solution is left for future research.

5 . Discussion.

In this paper we have developed a new variational scheme for approximating the spatial prob- ability distribution. The replica symmetric solutioii gives an excellent approximation to the numerical d a t a for high temperatures f > 1. For the tail of the spatial probability it predicts a Gaussian decay with a linear exponential correction. At lower temperature, the fit in tail region is not that good, either because the nnrnrrical h t a . i s insufficient and one needs to average over a higher number of realizations, or a solution with replica symmetry breaking is needed (or both). Of course since we are dealing after all with a n approximation, we are never guaranteed a perfect fit. For all temperatures tested the present method gives a much better fit to t h e probability distribution than using the Hamiltonian variational method, b o t h with or without RSB.

When the dat.a for 5? = 0.23 is displayed in a log-log plot, [16] a crossover is observed from a behavior P

-

-

P(w)

-

is depicted as open triangles in figure 3 for the range 6 < w 2 < 16. Since in this case we are not really observing t h e tail region (according to t h e definition given in Eq. (32)), this behavior

N O 1 1 MANIFOLDS IN RANDOM MEDIA 1593

may be a transient or intermediate behavior. We should also be cautious about the data in this region, because as mentioned in the last section it is possible that more realizations are needed.

On the other hand, if we do take the cubic w dependence of the log of the probability distribution seriously, this immediately rings a bell because of sonie work done by Villain e t al.

[l] on the toy model. What they actually showed was that the probability for rare realizations of the disorder which make H ( w )

-

-

-

for values of w3 >> 2 g / f i 2 (I have transformed their notation to ours). This could lead to the conclusion that the spatial probability distribution should also behave like a cubic power of U. This is true only if the probability distribution is completely dominated hy the rare realizations of this kind, a fact that was never claimed by Villain et al.. It is certainly possible that other realizations which give lower values to exp(-PH) but are nonetheless more abundant dominate the average value. Villain's argument is valid for any temperature. Our numerical results show that one certainly does not get a cubic behavior of the probability distribution a t high temperature for values of w >> ( ~ / f i ~ ) ' / ~ . In that case the rare realizations of of the type considered by Villain still exist but. they do not dominate the probability distribution.

This is actually easy to understand. Let us define two different length scales in the problem.

The first, int.roduced by Villain et al. which we call is defined as

1 j 3

El 2

(3)

is the length above which the probability for rare events of magnitude 1 goes like A exp( - w3 -)

E:

(39)

with some undetermined constant A . The second length, which I introduced in equation ( 3 2 ) ,

is the Iength for which the asymptotic behavior in the tail starts according to our investigation in section 3. For w above this value we have found the behavior

P(LL) ^N exp(-81w2/2 + ^{g,8'Iwl i} ~ ( 0 ) ) , (42) where C(0) is given in equation (34). Since

We see that for > 1.6, the cubic behavior is never realized since the contribution of the rare realizations of the Villain type to the averaged probability is smaller than the behavior given by equation (42). This explain the perfect fit of the asymptotic behavior given in equation (37) and the numerical data, starting at a very low value of w.

On the other hand for T << 1.6 it is conceivable to have t,wo tail regimes, the first with

El < ^U < E2 in which the probability has the cubic behavior in w because it is dominated by rare configurations of the Villain type, and a second tail regime for w > ⁽² for which the asymptotic behavior is changed to a quadratic behavior given by equation (42), if the results of the RS solution are valid (or possibly to a different asymptotic beliavior which will emerge

1594 JOITRIiAL DE PHYSIQUE I N"11

from a RSB solut.ion). This is because one can easily check that at the border between these two regimes the two expressions given by equation (40) and equation (42) become c ( m p a a b k in magnitudc, and the quadrat,ic behavior wins for w > c2. ^As the (reduced) teniperature is lowered from 1 the first regime (cubic beliaviorj is expected to grow iii size and include all of the tail at T = 0 . Siniulations of directed polymers a t zero temperat.nre. which can be =lapped int.0 corresporiding results for the toy-model [9]; showed the beanning of a change in the form of the spatial probability distribution from

-

-

There is some difficulty though, to explain the apparent, cubic behavior in figure 3 purely in terms of Villain's coiifi,qirati(iiis because we simulated "only" over 106 realizations and thus if the average is dominated by a siugle realization whose contribution is unity, the value of the average should be of the order of 10-G E esp(-14) which far exceeds the value of the distributiiin in most of this region (except a t the very beginning/. One might argue that there are ot,her realizations with somewhat smaller contributions to the average t h a n Villain's which also gives rise to a similar cubic behavior but this needs to be verified.

We hope that this work will stimulate further iuvestigation of the behavior of the tail of the probability distribution at low temperatures. Three important questions that need a definite answer are

1. Is there a RSB sulutiori tt3 our new variational equations?

2 . Is a cubic dependence of the log of the probability indeed realized over a large region when the temperature is very low?

3. Is the asymptotic behavior given by equation (34) which works so well for high temperatures when i ~ ) > E2. also valid asyniptotically at low temperatures'?

We also hope that it will be possiblc to extend the variational method developed in this paper for the zero-dimmsional nianifold directly to higher dimerisioual manifolds in razidom media.

Acknowledgments.

I tharik Professor C . Parisi for instigat,ing the current investigation, and for his k i d Iiospi- tality a t the Uiiiversity of Rome. where this research has beguu Support of the Natioual Science Fouridation urider grant number D?dR-9016907 is gratefully acknowledged. We also ackiiowledge Cray time allocation from t,he Pittsburgh Supercoinputer cent,er.

Appendix A.

Iii this Appendix me derive equatiou (24) saarting from equation (23). Let us shift the variables w, in equation (23)

lda

-

such that the liuear term in wl in the exponential is eliminated. This is achieved by choosing

A, = -(i/DjGIak (A.2)

KO11 M.'liYIFOI,DS IN RANDOM 3tEDI.A 1595

We now expand the function f iii a power series ahout 0. (this is used only as a. tool derive our result. It may not he necessary for f t,o be analytic about 0)

We w e the following formula to integrate over w i . . . un:

dwi...dw, (wa - L C ~ ) ~ ' exp =

(-4.5)

m(ip ^{i a )}

J

: (h/fi)"/'((Det G)'/' ~ ~ ~ ~ ( 1 , 2 ) (Gas + ^{Gab -} 2G-b)" ,

provided .r is a non-negatire integer. For s being half-int,eger the integral is 0. Wc obt,ain

1596 JOURNAL DE PHYSIQUE I NO11

Using this representation for f ( z , y ) t h e integral over I; in equation (A.6) can be performed, a n d we o b t a i n t h e desired result given in equation (24).

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[Z] Huse D.A. and Henley C.L., Phys. Rev. Lett. 54 (1985) 2708;

Iiusc D.A., Henley C.L. and Fisher D.S.; Phys. Rev. Lett. 55 (1985) 2924.

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[ I l l Rouchauld J.P., Mezard M. and Yedidia J.S., Phys. Rev. L e t t . 67 (1991) 3840.

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