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

Abstract

Let $u,v \in \mathbb{R}^\Omega_+$ be positive unit vectors and $S\in\mathbb{R}^{\Omega\times\Omega}_+$ be a symmetric substochastic matrix. For an integer $t\ge 0$, let $m_t = \smash{\left\langle v,S^tu\right\rangle}$, 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 $m_{t+2} \ge m_t^{1+2/t}\!,$ an inequality studied earlier by Blakley and Dixon (also Erd\H{o}s and Simonovits) for $u=v$ and shown true under the restriction $m_t\ge e^{-4t}$. Moreover we prove that for any $\epsilon>0$, a stronger inequality $m_{t+2} \ge t^{1-\epsilon}\cdot \smash{m_t^{1+2/t}}$ holds unless $m_{t+2}m_{t-2}\ge \delta m_t^2$ for some $\delta$ that depends on $\epsilon$ only. Phrased differently, $\forall \epsilon> 0, \exists \delta> 0$ such that $\forall S,u,v$ \begin{equation*} \frac{m_{t+2}}{m_{t}^{1+2/t}}\ge \min\left\{t^{1-\epsilon}, \delta\frac{m_t^{1-2/t}}{m_{t-2}}\right\}, \quad \forall t \ge 2, \end{equation*} 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 $\exp\left(\Omega(k\log k)\right)$.