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Symmetries of non-linear ODEs: lambda extensions of the Ising correlations

S. BoukraaJ. -M. Maillard
Dec 2022
This paper provides several illustrations of the numerous remarkableproperties of the lambda-extensions of the two-point correlation functions ofthe Ising model, sheding some light on the non-linear ODEs of the Painlev\'etype. We first show that this concept also exists for the factors of thetwo-point correlation functions focusing, for pedagogical reasons, on twoexamples namely C(0,5) and C(2,5) at $\nu = -k$. We then display, in alearn-by-example approach, some of the puzzling properties and structures ofthese lambda-extensions: for an infinite set of (algebraic) values of $\lambda$ these power series become algebraic functions, and for a finite set of(rational) values of lambda they become D-finite functions, more preciselypolynomials (of different degrees) in the complete elliptic integrals of thefirst and second kind K and E. For generic values of $ \lambda$ these powerseries are not D-finite, they are differentially algebraic. For an infinitenumber of other (rational) values of $ \lambda$ these power series are globallybounded series, thus providing an example of an infinite number of globallybounded differentially algebraic series. Finally, taking the example of aproduct of two diagonal two-point correlation functions, we suggest that manymore families of non-linear ODEs of the Painlev\'e type remain to be discoveredon the two-dimensional Ising model, as well as their structures, and inparticular their associated lambda extensions. The question of their possiblereduction, after complicated transformations, to Okamoto sigma forms ofPainlev\'e VI remains an extremely difficult challenge.