4 research outputs found

    Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations

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    We give the exact expressions of the partial susceptibilities χd(3)\chi^{(3)}_d and χd(4)\chi^{(4)}_d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, 3F2([1/3,2/3,3/2], [1,1]; z)_3F_2([1/3,2/3,3/2],\, [1,1];\, z) and 4F3([1/2,1/2,1/2,1/2], [1,1,1]; z)_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z) hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for χd(3)\chi^{(3)}_d and χd(4)\chi^{(4)}_d. We also give new results for χd(5)\chi^{(5)}_d. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the nn-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) q+1Fq_{q+1}F_q hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page

    Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

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    International audienceWe study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z2\mathbb{Z}^2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1-1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N2\mathbb{N}^2, counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201--215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19×419 \times 4 combinatorially meaningful specializations only four are algebraic functions

    2-descent for second order linear differential equations

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