Geometric complexity theory is an approach towards the separation of fundamental algebraic complexity classes. Two papers by Mulmuley and Sohoni [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496-526; SIAM J. Comput., 38 (2008), pp. 1175-1206] pursue this goal via representation theoretic multiplicities in coordinate rings of specific group varieties. The papers also conjecture that the vanishing behavior of these multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions was recently disproved in 2016 in two successive papers, [C. Ikenmeyer and G. Panova, Adv. Math., 319 (2017), pp. 40-66] and [P. Burgisser, C. Ikenmeyer, and G. Panova, J. Amer. Math. Soc., 32 (2019), pp. 163-193]. This raises the question of whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. This paper provides for the first time a setting where separating with multiplicities can be achieved, while separation with occurrences is provably impossible. Our setting is surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e., a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite's reciprocity theorem, which proves Foulkes's conjecture for a new infinite family of cases.

We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (Pemantle's Min-Plus Binary Tree, 2017.[math.PR]), and to the critical random hierarchical lattice studied by Hambly and Jordan (Adv Appl Probab 36(3):824-838, 2004. 10.1239/aap/1093962236). We prove distributional convergence for the processes, after rescaling, by showing that their evolutions can be understood as a discrete analogues of certain convection-diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.