Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees

In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions $\operatorname {PD}(\alpha,\theta)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and $\mathrm {Coag}_{\alpha,\theta}$, respectively, with the following property: if the input to $\mathrm {Frag}_{\alpha}$ has $\operatorname {PD}(\alpha,\theta)$ distribution, then the output has $\operatorname {PD}(\alpha,\theta+1)$ distribution, while the reverse is true for $\mathrm {Coag}_{\alpha,\theta}$. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between $\operatorname {PD}(\alpha,\theta)$ and $\operatorname {PD}(\alpha\beta,\theta)$. Repeated application of the $\mathrm {Frag}_{\alpha}$ operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation--fragmentation duality.Comment: Published at http://dx.doi.org/10.1214/105051606000000655 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Second Quantization and the Spectral Action

We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $\mu$ approaches $0,$ can be expressed in terms of the Riemann zeta function. This recovers a result of Chamseddine-Connes-van Suijlekom.Comment: Author list is expanded. The calculations in the new version are extended to two more Hamiltonians. New references adde