Block spin transformation

Definition

Let $d \ge 1$, $L \ge 1$, and let $b \ge 3$ be an odd integer. Let $\Lambda = \set{0, 1, \dots, bL - 1}^d$ and $\Lambda' = \set{0, 1, \dots, L - 1}^d$. For $y \in \Lambda'$, the block of $y$ is $B_y = by + \set{0, 1, \dots, b - 1}^d \subset \Lambda$. The blocks $B_y$, $y \in \Lambda'$, partition $\Lambda$, and each has $b^d$ sites, an odd number. The majority-rule block spin transformation is the map $T \colon \set{-1, 1}^\Lambda \to \set{-1, 1}^{\Lambda'}$ given by $$(T\sigma)(y) = \sgn\Big(\sum_{x \in B_y} \sigma(x)\Big), \qquad y \in \Lambda'.$$ Let $H \colon \set{-1, 1}^\Lambda \to \R$ be a Hamiltonian, which includes the inverse temperature. The renormalized Hamiltonian $H' \colon \set{-1, 1}^{\Lambda'} \to \R$ is defined by $$e^{-H'(\sigma')} = \sum_{\sigma \in \set{-1, 1}^\Lambda : \, T\sigma = \sigma'} e^{-H(\sigma)}.$$