A Lower Bound for the Mass of a Random Gaussian Lattice*

Random mass gaussian lattices are lattice systems where the single site distribution has the form ( i da(a)e-a~2)d~). d4 An example is ~-~-~. Related systems have been discussed quite frequently, at least in one dimension [1]. 1 Let da(a) be a Borel measure on (0, co) such that da(a) (1 + a)1/2 < oo. (1) For/~ > 0, define dmu(~) = (~ da(a) e{~ + ") ~2) d~. (2) Let Loo CIR a be a unit lattice centered on the origin, parallel to the coordinate axes. d L denotes the finite part of L~ contained in the box I~ [ lj + 1/2, l j t /2] where j = l (Ij) are given integers. On the space IR Lq, where tLI denotes the number of lattice points in L, define the probability measure _ 1 I ] dmu(~h) e(4''aD°) , (3) dPL, ~ -ZL," l e L (4, AD(b) = ~ (~b;@)2. (4) l , l ' * Supported by N.S.F. Grants PHY 76-17191, MPS 10751 1 The thermodynamic limit is taken after integrating over the masses, in this paper O010-3616/78/0062/0079/]101.00 80 D. Brydges and P. Federbush Z L , • is the normatisation. A n is the finite difference laplacian with Dirichlet boundary conditions, so the sum in (4) is over nearest neighbor lattice points in L~ and qS~-0 if l~L. The measure for the random mass gaussian lattice is to be obtained by taking limits L / " L ~, #'-~o in that order. By Griffith inequalities the moments ofdPL, ~ are monotone increasing as ILl increases and # decreases, therefore existence reduces to uniform upper bounds. In order to state and prove the theorem, we define for n > o, dm~,n(d?) = (~ da(a) ( 2 d 2 d + a)~ e-(~+ u)4~2) d¢ , (5) ZT,.,. = ~ l-[ dm,,,(Ol) e(¢'~4~). (6) l ~ T T is a lattice wrapped around a torus. Given L, it is defined by identifying the d boundary points of L~c~ 1~ I I s 1 / 2 , l~+1/2] in the obvious way. z~ w is the j = l finite difference laplacian with periodic boundary conditions defined by an equation like (4) in which l,l' are summed over nearest neighbors in T. Corresponding to dPL, . is a measure dPw, . obtained by replacing L by T, A D by A T and ZL, u by ZT,,, o. The periodic pressure is defined by Pu,,= LiimL~o JTt1 logZr,u,," (7) Theorem. The two point function lim lim ~ dP L ~ (~¢v #-~0 L L~ exists and is 0(e-MlZ-q) as tl l'l ~ oo for some M > 0 provided A l i m i n f ( P o P i )>0 ( M > A ) . u ~ O #' ~' " Remarks. The inequality A >0 is an obvious consequence of the definitions. We think A >0 will hold for d > 3, provided da(a)# 6(a). For d < 3, one must either place additional restrictions on dc~(a) near a =0 to ensure even existence as # ~ 0 or look at correlations of different quantities such as grad~b. The proof will use the following proposition ~ which may also be of interest. Proposition. Let b = (bz) be a strictly positive function on T Then (b A r)~r ~ = ~ l~ (2d + bz) ,(z,,o) co I~T co is summed over all random walks on T o f arbitrarily many nearest neighbor steps starting at l, ending at l'. n(l, co) is the number of times co hits I. The left hand side means the I, I' entry of the matrix inverse. Proof o f Proposition. (A T A.) (b A)-~ = (b + 2 d 2 d A)~, (8) =(b+2d) -~+(b+2d) ~(2d+A)(b+2d) -~ + . . . . (9) i This is a re formula t ion of a well known theo rem in r a n d o m walk Random Gaussian Lattice 81 This is the resolvent expansion in the off diagonal elements. The last line can be rewritten as the right hand side of the proposition because (2d)-l(2d+A) generates random walk. (It is a matrix with positive elements which sum to one,) Proof of Theorem by a Griffiths inequality 0 <= ~ dPL.u4~ r < ~ dPr,.4t~,,. (10) Substitute for dm~,(~) using (2). The right hand side becomes (Z-ZT,~,0, A---AT) i /2Z-1 ~ lq da(G) det1/2(2(//qa A)) (//+ a A)t_r 1 , (11) k e T =1/2z-1Y 1~ da(ak)d~kexp{--(//+ak)(°2}(//+a--A)l~t ~e(4''a4)" (12) k e T Therefore by the proposition dPT,~,~t~e = 1/2 Z ~ (2d)-"(k'03) 03 k e T ' Z-1 ~ ]-I dmu,.(k,o))(~t)e(~'A4') " (13) k e T We now apply Osterwalder-Schrader positivity in the form of the chess board estimate [2] (Lemma 4.5) to show dmu,n(t,~)((al) " l e T 1 < i-I (ZT,u,.(z,o,)) IT/" (14) l e T Combine (10), (13), and (14) and pass to the limit L-~L~ using definition (7). lim ~dPL,u~l~ ~, ~ 1/2 Z I ] (2d) -"(~'03) L ~ L ~ 03 IELoo • exp (P~,,(t,03) P,,o) < conste At~-z't. (15) The last inequality is using the fact that each c9 must visit at least II-l ' t lattice points and P~,,>P,,1 for n > 1. Proof concluded. Remarks. (1) Representations like (13) can be obtained for n point functions. (2) By using 5(q5 2 1) = 1 [ daei~(4,~_ 1) (16) 2~z a representations like (13) can be obtained for rotators with n components. Despite the complex numbers in (13) one still obtains positive measures dmu,.((o). Acknowledgement. We would like to thank Tom Spencer and Barry Simon for valuable conversations, 82D. Brydges and P. Federbush References1. Lieb, E., Mattis, D.: Mathematical physics in one dimension, pp. t19--196. New York: AcademicPress 19672. Fr6hlich,J. : Phase transitions, Goldstone bosons, and topological superselection rules. Acta Phys.Austriaca, Suppl. XV, 133 (1976) Communicated by A. Jaffe Received March 29, 1978; in revised form May 8, 1978