Exact results for the extreme Thouless effect in a model of network dynamics

¹ Center for Soft Matter and Biological Physics, Department of Physics, Virginia Polytechnic Institute and State University - Blacksburg, VA 24061, USA
² Department of Physics, University of Houston - Houston, TX 77204, USA
³ Texas Center for Superconductivity, University of Houston - Houston, TX 77204, USA
⁴ Department of Mathematics, University of Houston - Houston, TX 77204, USA

Received: 5 September 2018
Accepted: 18 December 2018

Abstract

If a system undergoing phase transitions exhibits some characteristics of both first and second order, it is said to be of “mixed order” or to display the Thouless effect. Such a transition is present in a simple model of a dynamic social network, in which extreme introverts/extroverts always cut/add random links. In particular, simulations showed that , the average fraction of cross-links between the two groups (which serves as an “order parameter” here), jumps dramatically when crosses the “critical point” , as in typical first-order transitions. Yet, at criticality, there is no phase co-existence, but the fluctuations of f are much larger than in typical second-order transitions. Indeed, it was conjectured that, in the thermodynamic limit, both the jump and the fluctuations become maximal, so that the system is said to display an “extreme Thouless effect”. While earlier theories are partially successful, we provide a mean-field–like approach that accounts for all known simulation data and validates the conjecture. Moreover, for the critical system , an analytic expression for the mesa-like stationary distribution, P(f), shows that it is essentially flat in a range , with . Numerical evaluations of f₀ provide excellent agreement with simulation data for . For large L, we find , though this behavior begins to set in only for L > 10¹⁰⁰. For accessible values of L, we provide a transcendental equation for an approximate f₀ which agrees with data to better than ∼1% down to L = 100. We conjecture how this approach might be used to attack other systems displaying an extreme Thouless effect.