1. Random permutations with cycle weights

Random permutations have been a hot topic in probability theory and applied mathematics because of their connections to random matrices and other systems. See for instance the review article of A. Okounkov, the 2006 Fields medallist [13]. Impressive and detailed results have been obtained. However, mathematicians have almost always restricted themselves to the uniform distribution.

The model of interest involves weights that depend on the cycle lengths. Let _n be the symmetric group of permutations of n elements, and let r_j = r_j(π) denote the number of cycles of length j in the permutation π _n. We consider the probability

Here, α₁,α₂,… are given coefficients and h_n is a suitable normalisation. We have h_n = 1 in the uniform case, i.e. α_j ≡ 0. There is a “gauge symmetry”, namely α_jα_j + cj, that leaves the probability invariant, so that α_j = cj is equivalent to the uniform distribution. The case α_j ≡ c is known as the Ewens distribution.

Preliminary investigations suggest that this probability is asymptotically equal to the uniform distribution when α_j → 0 sufficiently fast as j →∞ (such that ∑ _j < ∞), see [2]. In the regime α_j = j log j, it seems that almost all indices belong to cycles of length log n. On the other hand, if α_j = -j^δ with δ > 1, the distribution concentrates on cyclic permutations (with a single cycle of length n).

We have investigated this model with Volker Betz and Yvan Velenik [4], and we obtained the following results in the particular but interesting case α_j = j^γ:

• The case γ < 0 is a special case of the model studied in [2], which is close to the uniform distribution.
• In the case γ = 0, i.e. when θ_j → θ (the Ewens case, asymptotically), we find that P(ℓ₁ > sn) → (1 - s)^θ. Thus, almost all indices belong to cycles whose length is a fraction of n.
• The case 0 < γ < 1 is surprising. At first glance we might expect smaller cycles than in the uniform case α_j ≡ 0. However, we find that almost all indices belong to a single giant cycle! We also prove two additional results: (i) there is a uniformly positive probability that all indices belong to a single cycle of lenth n; (ii) in some cases, finite cycles exist with uniformly positive probability.
• The case γ = 1 actually corresponds to uniform permutations.
• When γ > 1 the cycles become shorter, and ℓ₁ behaves asymptotically as ( log n)^1∕γ.

In the case of the Ewens distribution, Lugo obtained results independently, and a bit earlier [11].

2. Spatial random permutations

Let us turn to random permutations with a spatial structure. Permutations now involve points in the space and there is an extra feature, namely the points are penalised if they travel far. Thus our second model is the quenched model of spatial permutations. Let x = (x₁,x₂,…) be fixed points in ^d (they may form a lattice, or they may be chosen according to some point process). The probability of a permutation π _n is defined as

Here, ξ is a spherically symmetric, increasing function ξ : ^d → ∪{∞}. The model is illustrated in Fig. 1. It is motivated by the Feynman-Kac approach to Bose-Einstein condensation and it was progressively introduced in [19, 8, 1].

Figure 1: Illustration for a random set of points x, and for a permutation π on x. Isolated points are sent onto themselves.

The most probable permutation is the identity permutation — all points keep their positions. But there is an “energy-entropy competition” since there are many more permutations with non-trivial jumps. Numerical experiments have been realised when the points form a cubic lattice, and with the jump function ξ(x) = |x|². Here, β is the inverse temperature of the model — the factor finds its justification in the Feynman-Kac representation of quantum systems, see Section 3. The coefficients α_j were always zero. We observed in particular that infinite cycles occur at low temperature in dimension d ≥ 3, see [8].

Here are some of the main questions for this model.

• Establish the existence of infinite cycles when x is the realisation of a suitable point process.
• Show that the cycle distribution of permutations are the same for almost all x.
• Discuss the existence of a measure on infinite permutations.
• What happens when the coefficients α_j differ from zero?

Let us turn to the annealed model of spatial permutations. It is similar to the quenched model, but it includes an average over point locations. Let Λ be a bounded open subset of ^d. Then

This setting is closely related to the quantum Bose gas, see below. One is interested in the limit of large systems, n,|Λ|→∞, while keeping the density ρ = n∕|Λ| fixed. The natural parameter here is the density: as the density increases, particles have more and more neighbours and they can jump more easily; this results in longer cycles. The question is whether cycles become infinite above some finite critical density.

I studied this model with V. Betz. We showed in particular that macroscopic cycles occur under certain conditions on the function ξ and if the density is larger than a certain critical density. This was proved in [1] for α_j ≡ 0, and in [2] for α_j → 0 (such that ∑ _j < ∞). A remarkable aspect is that this critical density is given by an exact expression, that involves the Fourier transform of e-ξ. When e-ξ is compactly supported, or when it has sufficient decay, infinite cycles occur only in dimensions greater than 3. But we also obtained infinite cycles in dimension 1 and 2 for certain integrable functions e-ξ with slow decay. Our results extend earlier results of Sütö for the case ξ ~|x|², which corresponds to the ideal Bose gas [17]. We used techniques developed by Buffet and Pulé in [6].

Another model that involves space and permutations is the “random stirring model” introduced by Tóth for the study of the quantum Heisenberg model [18]. It has interesting probabilistic aspects, see for instance the study of Schramm [15]. It is about permutations of lattice sites and is therefore closely related to systems that are discussed here, even though the probability on permutations is a bit different. Efforts should be devoted to uncovering their relations.

3. Bose-Einstein condensation and infinite cycles.

Bosonic systems have fascinated people for a long time. The subject started in 1924–5 when Bose and Einstein noticed that a curious, genuinely quantum, phase transition occurs at low temperature. Einstein could even compute exactly the critical temperature!

Despite many interesting theoretical contributions, major questions remain open. Nobody has rigorously established the occurrence of a condensation in interacting systems (except for hard-core bosons on the cubic lattice, see Chapter 11 in [10] and references therein). Whether interactions enhance or discourage the condensation is not clear and is still currently debated, with conflicting reports in the literature. There is room for exciting new understanding, and for precise speculations leading to experimental verifications.

The Hamiltonian for a system of N quantum particles is the Schrödinger operator

that acts in the Hilbert space L_sym²(Λ^N) of symmetric complex-valued functions of N arguments. The operator Δ_i is the Laplacian for the particle i. Interactions are represented by a multiplication operator that involves the function U(x). We always suppose that U is nonnegative, spherically symmetric, and that it decays sufficiently fast to 0 as |x|→∞.

The key expression in statistical physics is the Gibbs operator e-βH. Using the Feynman-Kac formula, it is possible to represent the quantum system as a probability model. The bosonic nature of particles is responsible for the appearance of random permutations. In the case of the ideal Bose gas where particles do not interact, random permutations come with Hamiltonian

Bose-Einstein condensation should be related to the occurrence of infinite cycles. This was somehow suggested by Matsubara [12] and Feynman [7], and made more precise by Sütö in the case of the ideal gas [17]. See also [20, 21, 6] for clarification and some justification.

The situation is considerably more difficult in presence of interactions. The two-body interactions of the quantum particles translate into intricate multi-body interactions between permutation jumps. However, a computation has revealed that the correct interaction to lowest order is given by the following formula [22]:

This represents the interaction between the “jumps” xy and x′y′. Here, W_x,y^4β denotes the Wiener measure for the Brownian bridge between x and y in time 4β; U is the original interaction potential between quantum particles. The new Hamiltonian for permutations is

A major open question in this field is the effect of interactions on the Bose-Einstein condensation in dilute systems. The current consensus among physicists is that the change ΔT_c = T_c^(a) - T_c⁽⁰⁾ of the critical temperature behaves as

with positive constant c ≈ 1.3. Here, T_c^(a) denotes the critical temperature of a dilute homogeneous Bose gas and a is the scattering length of the (repulsive) potential that describes the particle interactions. These results are reviewed in Refs [16], where more details and additional references can be found. In this article with Robert Seiringer, we also propose a mathematically rigorous upper bound on the interacting critical temperature.

In a recent article with Volker Betz [3], we performed various calculations and simplifications. Those steps are not mathematically rigorous, but they are hopefully exact. We find a negative constant c = - ≈ -2.33, which implies that interactions decrease the critical temperature. Our findings contradict those of our physics colleagues. Since we trust both our method and the physics literature, we are left puzzled. The future will hopefully resolve these contradictions.

References