High-order Fuchsian equations for the square lattice Ising model: χ⁽⁶⁾

Abstract

We consider the Fuchsian linear differential equation obtained (modulo a prime) for˜χfor˜ for˜χ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of˜χof˜ of˜χ (1) and˜χand˜ and˜χ (3) can be removed from˜χfrom˜ from˜χ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms˜χterms˜ terms˜χ (3) and˜χand˜ and˜χ (4). We conjecture that a linear differential operator equivalent to a symmetric (n − 1)-th power of L E occurs as a left-most factor in the minimal order linear differential operators for all˜χall˜ all˜χ (n) 's.

Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?

“…The non-holonomic susceptibility series is known to be an infinite sum of holonomic functions [9,4], namely the n-fold integrals [7,8,10,11,12,41] χ (n) that are themselves diagonals of rational functions [20,29,42,43,44]. To some extent, the remarkable result [15] that the non-holonomic susceptibility series reduces to algebraic functions modulo 2 r can be seen as a property of these diagonal of rational functions, namely that these χ (n) reduce to zero modulo 2 r when n is large enough [24].…”

“…In contrast with linear differential equations one can have, for a given series, many non-linear differential equations of the same order, for instance N 2 = 0 and N 3 = 0. For linear differential equations, the (unique) minimal order linear ODE requires (paradoxically [12]) many more coefficients to be obtained from the so-called "guessing procedures" than higher order ODEs. When we use our (non-linear guessing) program it is not clear if the minimal order non-linear ODEs are the easiest to be obtained, requiring the minimal number of coefficients to be "guessed".…”

“…It was noted in a previous paper (see equations (47) and (48), in Section 5.2 of [24]), that the two diagonals of rational functions H 1 (x) and H 2 (x) (12) reduce, modulo a prime, to simple algebraic functions. For instance, modulo 7, the two diagonals of rational functions (12) identify with the series expansion of the following algebraic functions: is a solution of (13) where H 1 and H 2 are replaced by the two algebraic functions A 1 and A 2 , namely solution of…”

“…In a series of papers, Maillard and co-workers [7,8,9,10,11,12] found the linear ordinary differential equations (ODEs) satisfied by χ (3) , · · · , χ (6) , and showed how, with sufficient computer power, others could, in principle, be found. χ (1) is algebraic, and χ (2) satisfies a low-order ODE, but the order of the linear ODEs satisfied by χ (n) increases rapidly with n as does the degree of the polynomial coefficients of each derivative, so the number of series coefficients needed to discover the linear ODE also increases rapidly with n. In 2011, Chan et al [13] extended the work of Orrick et al to other two-dimensional lattices, and gave an expansion of the scaled form of the susceptibility to unprecedented accuracy.…”

On Christol’s conjecture

“…There is a plethora of multiple integrals in physics: Feynman integrals, lattice Green functions, the summands of the magnetic susceptibility of the 2D Ising model [2,22], that have very specific mathematical properties. These functions are D-finite, i.e., solutions of linear differential operators with polynomial coefficients, and have series expansions with integer coefficients.…”