Escribiendo urbanismos sociales: a propósito del trabajo de Andy Merrifield

El desconcierto invadirá al lector impróvido que se aproxime al trabajo de Andy Merrifi eld sin noticia de su periplo existencial y, especialmente, de su deriva vital en la última década: varios libros de teoría urbanística crítica, dos análisis, entre lo biográfi co y lo fi lológico, de pensadores franceses de la segunda mitad del siglo XX, la crónica de un retiro rural simbolizada en la fi gura de un asno y un «cuento de hadas dialéctico» sobre las miserias y futuros de la izquierda radical. Desde luego hay hilos comunes en estas intervenciones —la ciudad, el marxismo, la refl exión sobre ambos y su rechazo en sus formas actuales—, pero no son fáciles de advertir y, menos aún, de componer o explicar

Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed