content_image/362035.jpg content_image/362036.jpg content_image/362037.jpg content_image/362038.jpg
of ¯ ζ$_{t}$ is, 

<img src='content_image/362035.jpg'>

Unlike in the case where K = 2, the second term in the above variance expression does not reduce to a constant. We can show, however, using a procedure similar to Guibas and Veach [6, 7], that it is bounded by a constant: 

Thus, if we find a set of functions for α$_{k}$ that minimize the first term in σ [ ¯ ζ$_{t}$ ], no other choices for α$_{k}$ can decrease the variance by more than max ( 1 N$_{T}$$_{k}$ + 1 N$_{π}$$_{k}$ ) . 

Minimization of the first term in Eq. 9 by Lagrange multipliers leads to an optimized set of α$_{k}$ analogous to Eq. 4. As with K = 2, substitution of this α$_{k}$ leads to an estimator for the integral which can either be solved self-consistently or by finding the zero of an implicit function, 

<img src='content_image/362036.jpg'>

<img src='content_image/362037.jpg'>

<img src='content_image/362038.jpg'>

The corresponding variance may be estimated by substituting the appropriate α$_{k}$ into Eq. 9. 

The original and optimized forms of the replica gas identity were demonstrated by esti- mating the partition function of a two-dimensional 32 × 32 spin Ising model, using parame- ters similar to those previously described [3]. As in a typical Ising model simulation without 

