The q-deformed conformal matrix model partition function, after a Fourier transform, is a Toda tau-function whose shifted ratios satisfy the discrete Painleve q-PVI equation, with the string equation supplied by Virasoro constraints.
On the Continuum Limit of the Conformal Matrix Models
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abstract
The double scaling limit of a new class of the multi-matrix models proposed in \cite{MMM91}, which possess the $W$-symmetry at the discrete level, is investigated in details. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the paper \cite{FKN91a} is proposed, the corresponding partition functions being compared. All calculations are demonstrated in full in the first non-trivial case of $W^{(3)}$-constraints.
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Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
The q-deformed conformal matrix model partition function, after a Fourier transform, is a Toda tau-function whose shifted ratios satisfy the discrete Painleve q-PVI equation, with the string equation supplied by Virasoro constraints.