Toda Lattice Hierarchy and Zamolodchikov’s Conjecture

where the domain of integration for the operator is (0,∞). The quantity e(p) does not depend on tn’s. It should be remarked that our tn corresponds to tn/2 of refs.[1, 2]. Independently, Bernard and LeClair showed that φ(t) solves the shG equation[3]. Quite recently, Tracy and Widom proved that above φ(t) satisfies both the mKdV hierarchy and the shG hierarchy[2]. Their proof is based on the fact that the quantity φ(t) is expressed in terms of Fredholm determinant. In this letter, we will show that the Fredholm determinant D(λ; t) is nothing but a special case of τ -function for the Toda lattice hierarchy[4]. As in the appendix D of ref.[3], we start with finite domain of integration (0, ν) for ν finite, rather than infinite domain (0,∞). We then consider the quantity,