Near log-convexity of measured heat in (discrete) time and consequences

Let u, v is an element of R-+(Omega) be positive unit vectors and S is an element of R-+(Omega) (x) (Omega) be a symmetric substochastic matrix. For an integer t >= 0, let mt = (v, S(t)u), which we view as the heat measured by v after an initial heat configuration u is let to diffuse for t time steps according to S. Since S is entropy improving, one may intuit that m(t) should not change too rapidly over time. We give the following formalizations of this intuition. We prove that m(t)+2 >= m(t)(1+2/t), an inequality studied earlier by Blakley and Dixon (also Erdos and Simonovits) for u = v and shown true under the restriction m(t) > e(-4t). Moreover we prove that for any epsilon > 0, a stronger inequality m(t)+2 >= t(1-epsilon) . m(t)(1+2/)t holds unless m(t)+2m(t-2) >= delta m(t)(2) for some delta that depends on epsilon only. Phrased differently, for all epsilon > 0, there exists delta > 0 such that for all S, u, v m(t) broken vertical bar 2/m(t)(1+2/)t >= min {t(1-epsilon,) delta m(t)(1-2/t/)m(t-2},) for all t >= 2 which can be viewed as a truncated log-convexity statement. Using this inequality, we answer two related open questions in complexity theory: Any property tester for k-linearity requires Omega(k log k) queries and the randomized communication complexity of the k Hamming distance problem is Omega(k log k). Further we show that any randomized parity decision tree computing k-Hamming weight has size exp (Omega(k log k)).