自动机与正方格伊辛模型模素数幂的磁化率
Automata and the susceptibility of the square lattice Ising model modulo powers of primes
摘要：
研究了素数模幂的伊辛模型的全磁化率.我们找到确切的功能方程的全部磁化率模这些素数。回顾离散有限自动机的一些鲜为人知的结果，我们表明，这些结果可以被看作是一个事实的后果，即模2^r，一个人不能区分全磁化率从一些简单的对角线的有理函数，减少到代数函数模2^r，因此，满足确切的功能方程模2^r。我们勾勒出一个可能的物理解释，这些功能方程模2^r作为减少的主功能方程对应于无限阶对称，如椭圆曲线的同构。一个相关的例子是兰登变换，它可以被视为伊辛模型重整化群的精确生成元。我们强调的重要性，研究一类新的功能对应的比率的对角线的有理函数：他们减少到代数函数模幂的素数，他们可能有解决方案与自然边界。
英文摘要：
We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2^r, one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2^r, and, consequently, satisfy exact functional equations modulo 2^r. We sketch a possible physical interpretation of these functional equations modulo 2^r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.
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