  An important conjecture in additive combinatorics posits that the partition
rank and analytic rank of tensors are equal up to a constant, over any finite
field. We prove the conjecture up to logarithmic factors. Our proof is largely
independent of previous work, utilizing recursively constructed polynomial
identities and random walks on zero sets of polynomials. We also introduce a
new, vector-valued notion of tensor rank (``local rank''), which serves as a
bridge between partition and analytic rank and which may be of independent
interest.
