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We study the growth rate of the hard squares lattice gas, equivalent to thenumber of independent sets on the square lattice, and two related models -non-attacking kings and read-write isolated memory. We use an assortment oftechniques from combinatorics, statistical mechanics and linear algebra toprove upper bounds on these growth rates. We start from Calkin and Wilf'stransfer matrix eigenvalue bound, then bound that with the Collatz-Wielandtformula from linear algebra. To obtain an approximate eigenvector, we use anansatz from Baxter's corner transfer matrix formalism, optimised with Nishinoand Okunishi's corner transfer matrix renormalisation group method. Thisresults in an upper bound algorithm which no longer requires exponential memoryand so is much faster to calculate than a direct evaluation of the Calkin-Wilfbound. Furthermore, it is extremely parallelisable and so allows us to makedramatic improvements to the previous best known upper bounds. In all cases wereduce the gap between upper and lower bounds by 4-6 orders of magnitude.

Upper bounds on the growth rates of hard squares and related models via corner transfer matrices