OAI.ThorpNine.Harmonic.Thorp.LowPlanes.regular_trace_smoothing

The theorem states that there exists a real number P ≥ 2 such that, for every dimension d ≥ 1, two bounds hold. Here positions are the 2^d points of {0,1}^d (Boolean vectors of length d), and a random walk on the symmetric group of these positions is built from independent steps: a step with coin ξ, a Boolean function on length-(d−1) vectors, is the composition of a cyclic rotation of the coordinates with a switch that flips the first coordinate of x exactly when ξ applied to the remaining coordinates is true; run d t is the product of t such steps, with the latest step multiplied on the left, and law d t g is the probability that it equals g under uniformly random independent coins. For each Young diagram μ with 2^d cells, the Specht-type space spanned by the orbit of a polytabloid is a unitary representation of the symmetric group on the positions, and Q_μ is its average over d independent coin choices of the representing operator of run d d. The regular trace is the sum over all such μ of dim(V_μ) times the real part of the trace of |Q_μ|^P, where |A| = sqrt(A*A). The first bound is that this regular trace is at most 1 + (2^d)^(−10). The second is that for every integer v with P ≤ 2v and every permutation σ of the positions, the sweep distance, namely half the sum over permutations g of |law(g σ⁻¹) − 1/(2^d)!| for the walk of v·d steps (using the natural-number part of v), is at most (1/2)(2^d)^(−5).