# (p.382) Appendix B Evaluation of the Laplace Integral Outside the Analyticity Strip

# (p.382) Appendix B Evaluation of the Laplace Integral Outside the Analyticity Strip

The microcanonical partition function in formula (2.76) can be expressed using the Laplace representation of the Dirac $\delta $ function as

with $\beta >0$; this is the expression given in (2.78), with $E=N\epsilon $. As explained in Section 2.5, we divide the integral in (B.1) into three intervals, defined by ${\lambda}_{I}<-\delta ,\text{}-\delta <{\lambda}_{I}<\delta $ and ${\lambda}_{I}>\delta $, respectively, with $0<\delta <\mathrm{\Delta}$. Here we show that the contribution to the integral in *λ* coming from values of ${\lambda}_{I}$ outside the strip, i.e. for values of ${\lambda}_{I}$ with $|{\lambda}_{I}|\text{}>\mathrm{\Delta}$, is exponentially small in *N*.

Let us then consider first the value of $Z(\lambda ,N)$ in the two external intervals, i.e. for $|{\lambda}_{I}|\text{}>\mathrm{\Delta}$. We have

We see that this expression is proportional to $\u27e8\text{exp}\left(-i{\lambda}_{I}H\right)\u27e9$, the canonical expectation value of $\text{exp}\left(-i{\lambda}_{I}H\right)$, which we expect to be exponentially small for large *N*. We confirm this expectation rewriting the last expression as

where we have used the expression of the canonical partition in terms of the microcanonical entropy as in Eq. (1.64). This is an integral with a large phase. For *N* going to infinity, its value will be determined by the value of the integrand for $\epsilon $ equal to the integration extremes and to the values of the possible nonanalyticities of $s(\epsilon )$, all denoted by ${\epsilon}_{k}$ (see, e.g. (Bender and Orszag 1978)). We then have

(p.383)
where *c*_{k} are coefficients that could in principle be determined. It is clear that the successive integration over ${\lambda}_{I}$ in any one of the two external intervals of integration will then give a vanishing contribution, due to the very large oscillations. We are then left with

which is the first equality given in Eq. (2.79).