Planckian axions in string theory

We argue that super-Planckian diameters of axion fundamental domains can arise in Calabi-Yau compactifications of string theory. In a theory with N axions theta(i), the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form -pi < Q(i) (j)theta(j) < pi. We compute the diameter of the fundamental domain in terms of the eigenvalues f(1)(2) <= ... <= f(N)(2) of the metric on field space, and also, crucially, the largest eigenvalue of (QQ(inverted perpendicular))(-1). At large N, QQ(inverted perpendicular) approaches a Wishart matrix, due to universality, and we show that the diameter is at least N f(N), exceeding the naive Pythagorean range by a factor > p N. This result is robust in the presence of P > N constraints, while for P = N the diameter is further enhanced by eigenvector delocalization to N-3/2 f(N). We directly verify our results in explicit Calabi-Yau compactifications of type IIB string theory. In the classic example with h(1,1) = 51 where parametrically controlled moduli stabilization was demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys f(N) approximate to 0.013 M-pl. The random matrix analysis then predicts, and we exhibit, axion diameters approximate to M-pl for the precise vacuum parameters found in [1]. Our results provide a framework for pursuing large-field axion in flation in well-understood flux vacua.