Kardar-Parisi-Zhang growth in ɛ dimensions and beyond

We examine anew the relationship of directed polymers in random media on traditional hypercubic versus hierarchical lattices, with the goal of understanding the dimensionality dependence of the essential scaling index β at the heart of the Kardar-Parisi-Zhang universality class. A seemingly accurate, but entirely empirical, ansatz due to Perlsman and Schwartz, proposed many years ago, can be put in proper context by anchoring the connection between these distinct lattice types at vanishing dimensionality. We graft together complementary perturbative field-theoretic and nonperturbative real-space renormalization group tools to establish the necessary connection, thereby elucidating the central mystery underlying the ansatz's uncanny apparent success, but also revealing its intrinsic limitations. Furthermore, we perform an extensive Euler integration of the KPZ equation in 3+1 dimensions which, bolstered by a separate directed polymer simulation, allows us an estimate for the critical exponent β_{3+1}^{KPZ}=0.1845(4) that greatly improves upon all previous Monte Carlo calculations in this regard and rules out the Perlsman-Schwartz value, 0.1882^{+}, in that dimension. Finally, leveraging this hybrid RG partnership permits us a versatile, more potent, tool to explore the general KPZ problem across dimensions, as well as a conjecture for its key critical exponent, β=1/2-0.22967ɛ, as ɛ→0, testable in a three-loop calculation.