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By C. Bardos, J. M. Lasry, M. Schatzman

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One of the basic problems in the context of statistical mechanics will be to control all this procedure with respect to the dimension. There is also an analog of the above theorem for a phase with more than one minimum and under suitable assumptions at oo, like the existence of the inequality *(a:)> 7 f N p - C » VzelR", 33 Classical Laplace method for some p > 0 and C > 0, one can also consider Laplace integrals over JRn. The assumption that $ is real can be made weaker if one assume holomorphy of a and $ and we then enter in the world of the saddle point method.

T J )n E M /4~ V r + 1 UR m My) My) dy? ) 1-dimensional lattices 45 and, taking the limit as n —>• oo, we finally obtain the following identity lim Cov(")(/«,/«) = V / f(y)uk(y)ui(y)dy J JR fc>l . 3) JR where var(/) = / ux(x)2 f(x)2 dx - ( f Ul(x) f(x) dx\ is the variance of / with respect to the measure on IR : u\{x) dx. For the first inequality, we have used the property that, by the Parseval equality, we have / (f(x)u1(x))2dx = '^2 ( / f(x)ui(x)ue(x)dx) , remembering that Uj is an orthonormal basis.

B2 . 8) • Step 2 Once we have determined t, we immediately get the corresponding H(t), by writing / exp -P(x, y) dy = ^i(i) . 9) JR We obtain Mi(«) = f e x p - ^ ± V dy = (-^~)i . 10) * a-\-t The eigenvector being strictly positive, it is necessarily the ground state (according to the Perron-Probenius Theorem), that is the eigenvector corresponding to the largest eigenvalue of K. We have consequently found the "first" eigenvalue and the corresponding eigenvector. JR • Step 3 We observe that u\(x) = {^)* exp—|a; 2 satisfies ui(a;)-Mxui(a;)=0.

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