We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL (N , C). The corresponding operator acts in the direct sum of N (n − 3) copies of L 2 S 1 . Its kernel has a block integrable form and is expressed in terms of fundamental solutions of n−2 elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant n-point system via a decomposition of the punctured sphere into pairs of pants. For N = 2 these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to n = 4 gives a series representation of the general solution to Painlevé VI equation.

We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.