Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 51 / 3 / a / Solution

A blocking kernel is a conditional probability $K({\sigma }^{\prime }\mid \sigma )$ for replacing the original spins by block spins. It must satisfyfor each original configuration. For example, divide the lattice into blocks of side $b$ and assign each block the sign of its total spin; a tie can be resolved with equal probabilities for the two signs. The product of these block rules is a normalized blocking kernel.

The real-space renormalization group defines the effective Boltzmann weight by summing the old weights against this kernel. Its normalization preserves the full partition function. The blocked lattice has spacing $ba$ and $N^{\prime }=N/b^D$ sites. Rescaling lengths restores the original lattice-spacing convention. An exact transformation generally produces every symmetry-allowed interaction, not just the nearest-neighbor term; retaining only a few couplings is a further approximation.