We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite @math -systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
ii) The ideal gas with the Haldane statistics and the Sutherland-Wu equation. The series @math has an interpretation of the grand partition function of the ideal gas with the Haldane exclusion statistics @cite_16 . The finite @math -system appeared in @cite_16 as the thermal equilibrium condition for the distribution functions of the same system. See also @cite_1 for another interpretation. The one variable case ) also appeared in @cite_26 as the thermal equilibrium condition for the distribution function of the Calogero-Sutherland model. As an application of our second formula in Theorem , we can quickly reproduce the cluster expansion formula'' in [Eq. (129)] I , which was originally calculated by the Lagrange inversion formula, as follows: where @math is the solution of ). The Sutherland-Wu equation also plays an important role for the conformal field theory spectra. (See @cite_23 and the references therein.)
@cite_23 @cite_16 @cite_1 @cite_26
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We discuss the relationship between the classical Lagrange theorem in mathematics and the quantum statistical mechanics and thermodynamics of an ideal gas of multispecies quasiparticles with mutual fractional exclusion statistics. First, we show that the thermodynamic potential and the density of the system are analytically expressed in terms of the language of generalized cluster expansions, where the cluster coefficients are determined from Wu’s functional relations for describing the distribution functions of mutual fractional exclusion statistics. Second, we generalize the classical Lagrange theorem for inverting the one complex variable functions to that for the multicomplex variable functions. Third, we explicitly obtain all the exact cluster coefficients by applying the generalized Lagrange theorem. @S0163-1829 98!03335-9# We derive an exact integral representation for the gr and partition function for an ideal gas with exclusion statistics. Using this we show how the Wu's equation for the exclusion statistics appears in the problem. This can be an alternative proof for the Wu's equation. We also discuss that singularities are related to the existence of a phase transition of the system. We study the properties of the conformal blocks of the conformal eld theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate elds, the conformal blocks obey second order dierential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with nontrivial braiding properties. A generalized duality property relates the two types of second order degenerate elds. By studying this duality we found that the excited states of the CalogeroSutherland Hamiltonian are characterized by two partitions, or in the case of WAk 1 theories by k partitions. By extending the conformal eld theories under consideration by a u(1) eld, we nd that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in generalk, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states. We consider the solution of the stochastic heat equation @TZ D 1 @ 2 X ZZ P W with delta function initial condition Z.T D0;X DiXD0 whose logarithm, with appropriate normalization, is the free energy of the con- tinuum directed polymer, or the Hopf-Cole solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions, the crossover distributions, which interpolate between a standard Gaussian dis- tribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest-descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion process with antishock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes. © 2010 Wiley Periodicals, Inc.