Painleve versus Fuchs

Abstract

The sigma form of the Painlev{é} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{é} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{é}.

18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 1905

References in corpus (1)

• Square lattice Ising model susceptibility: connection matrices and singular behavior of ${\chi }^{(3)}$ and ${\chi }^{(4)}$

Cited by in corpus (12)

• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Holonomy of the Ising model form factors
• A Fuchsian matrix differential equation for Selberg correlation integrals
• Form factor expansions in the 2D Ising model and Painlevé VI
• Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
• The anisotropic Ising correlations as elliptic integrals: duality and differential equations
• Fuchs versus Painlevé
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
• Scaling functions in the square Ising model
• The lambda extensions of the Ising correlation functions C(M, N)
• On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model