Bounds on Rate of Convergence for the Shuffled Discrete Heat Equation in Zd

We explore the effects of interleaved shuffling on the rate of convergence for the discrete heat equation with Dirichlet boundary conditions. We derive a closed form for the expected value of the shuffled discrete heat equation and establish bounds on its rate of convergence. In particular for any connected region D ⊂ Z with volume |D| and a non-negative initial state h0 ∈ R|D|, there is an upper bound on the spectral radius associated with the shuffled discrete heat equation that grows on the order of 1 − Ω(1/|D|). An analogous lower bound for the standard discrete heat equation is also derived which grows on the order of 1−O(1/|D|).