Abstract
We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models — non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf’s transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter’s corner transfer matrix formalism, optimised with Nishino and Okunishi’s corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude. Nous étudions le taux de croissance du système de particules dur sur un réseau carré. Ce taux est équivalent au nombre d’ensembles indépendants sur le réseau carré. Nous étudions également deux modèles qui lui sont reliés : les rois non-attaquants et la mémoire isolée d’écriture-réécriture. Nous utilisons techniques diverses issues de la combinatoire, de la mécanique statistique et de l’algèbre linéaire pour prouver des bornes supérieures sur ces taux de croissances. Nous partons de la borne de Calkin et Wilf sur les valeurs propres des matrices de transfert, que nous bornons à l’aide de la formule de Collatz-Wielandt issue de l’algèbre linéaire. Pour obtenir une valeur approchée d’un vecteur propre, nous utilisons un ansatz du formalisme de Baxter sur les matrices de transfert de coin, que nous optimisons avec la méthode de Nishino et Okunishi qui exploite ces matrices. Il en résulte un algorithme pour calculer la borne supérieure qui n’est plus exponentiel en mémoire et est ainsi beaucoup plus rapide qu’une évaluation directe de la borne de Calkin-Wilf. De plus, cet algorithme est extrêmement parallélisable et permet ainsi une nette amélioration des meilleurs bornes supérieures existantes. Dans tous les cas l’écart entre les bornes supérieures et inférieures s’en trouve réduit de 4 à 6 ordres de grandeur.
Highlights
We study the growth rate of the statistical mechanical model of a hard square lattice gas
To form the approximate eigenvector, we use corner transfer matrix formalism. This is a very powerful approach developed in statistical mechanics by Baxter [1, 2, 4] as a way to estimate the partition function of various lattice models, both numerically and via series expansions [4, 7, 8]
To optimise the choice of vector, we use an extension of CTM known as the corner transfer matrix renormalisation group (CTMRG) method, developed by Nishino and Okunishi [24, 25]
Summary
We study the growth rate of the statistical mechanical model of a hard square lattice gas. Almost all works which compute bounds on the growth rates of the hard squares and related models use the above two inequalities (two recent exceptions being [15] and previous work by the authors [9], both of which use methods from statistical physics). To form the approximate eigenvector, we use corner transfer matrix formalism This is a very powerful approach developed in statistical mechanics by Baxter [1, 2, 4] as a way to estimate the partition function of various lattice models, both numerically and via series expansions [4, 7, 8]. To optimise the choice of vector (and the associated auxiliary matrices), we use an extension of CTM known as the corner transfer matrix renormalisation group (CTMRG) method, developed by Nishino and Okunishi [24, 25]
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