Near log-convexity of measured heat in (discrete) time and consequences
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Let u,v ∈R^Ω_+ be positive unit vectors and S∈R^Ω×Ω_+ be a symmetric substochastic matrix. For an integer t> 0, let m_t = 〈 v,S^tu〉, 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 Erdős and Simonovits) for u=v and shown true under the restriction m_t> e^-4t. Moreover we prove that for any ϵ>0, a stronger inequality m_t+2> t^1-ϵ·m_t^1+2/t holds unless m_t+2m_t-2>δ m_t^2 for some δ that depends on ϵ only. Phrased differently, ∀ϵ> 0, ∃δ> 0 such that ∀ S,u,v m_t+2/m_t^1+2/t>{t^1-ϵ, δm_t^1-2/t/m_t-2}, ∀ 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 Ω(k k) queries and the randomized communication complexity of the k-Hamming distance problem is Ω(k k). Further we show that any randomized parity decision tree computing k-Hamming weight has size (Ω(k k)).
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