A one-dimensional model with phase transition

Abstract

Two repellent particles are bound to occupy two among thek _n +1 adjacent sites 0=x ⁽ⁿ⁾₀ <x ⁽ⁿ⁾₁ <...<x ⁽ⁿ⁾_kn =1, sayx ⁽ⁿ⁾_q ,x ⁽ⁿ⁾_q+1 . Define the Hamiltonian ℋ ⁽ⁿ⁾_q =−ln(x ⁽ⁿ⁾_q+1 −x ⁽ⁿ⁾_q ) and the partition function

We discuss the behaviour of the function

$$F(\beta ) = \mathop {\lim \sup }\limits_{n \to + \infty } \frac{{\ln Z(\beta , n)}}{{\ln k_n }},$$

closely related to the free energy. We prove that the smallest real zero ofF(β) is equal to the fractal dimension of the system and that this number, when less than one, is a critical value whereF is not analytic.