Asymptotic Statistics of Cycles in Surrogate-Spatial Permutations

Abstract

We propose an extension of the Ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group. This model is primarily motivated by a natural approximation to the so-called spatial random permutations recently studied by V. Betz and D. Ueltschi (hence the name "surrogatespatial"), but it is of substantial interest in its own right. We show that under the suitable (thermodynamic) limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles. Moreover, using a greater analytic tractability of the surrogate-spatial model, we obtain a number of new results about the asymptotic distribution of the cycle lengths (both small and large) in the full range of subcritical, critical, and supercritical domains. In particular, in the supercritical regime there is a parametric "phase transition" from the Poisson-Dirichlet limiting distribution of ordered cycles to the occurrence of a single giant cycle. Our techniques are based on the asymptotic analysis of the corresponding generating functions using Pólya's Enumeration Theorem and complex variable methods.

The number of cycles in random permutations without long cycles is asymptotically Gaussian

“…In particular, this implies that, contrary to the situation in the present paper, there is a positive fraction of points in finite cycles. The latter property motivates the name 'surrogate spatial random permutations' in the title of Bogachev and Zeindler (2015): while for random permutations it is in general rare to see a fraction of indices in finite cycles due to very strong entropic effects, this phenomenon becomes the norm when we add a spatial component to the model. These spatial random permutations are still relatively little understood, but appear to have many intriguing properties such as a phase transition from a regime without long cycles to one with a mixture of long and short cycles (proved for a special case in ) and a rich geometry of the set of points in long cycles (see Grosskinsky et al, 2012;Betz, 2014 for some simulations).…”

“…Firstly, Maples et al (2012) consider the generalized Ewens measure with cycle weights of the form j αcj (σ) with α > 0. Secondly, Bogachev and Zeindler (2015) replace the sequence α j by a doubly indexed sequence α j,n . Under suitable assumptions on the asymptotics of the α j,n , they show (among many other things) that the number of cycles is of order n and satisfies a central limit theorem.…”

Random permutations without macroscopic cycles

“…it depends only on the cycle structure. One variety of such models are those with cycle weights, including the Ewens model [17] with applications in genetics, or more general cycle weight models [9,11,15,16] with applications in quantum many body systems [7,8]. Another variant is to condition on the absence of cycles of a given length.…”

Loop Correlations in Random Wire Models

“…This conjecture has been rigorously established in a model of spatial permutations related to the quantum Bose gas [10,14,18]. This model has a peculiar structure that makes it possible to integrate out the spatial variables and to use tools from asymptotic analysis, so there were suspicions that this property was accidental.…”