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Register a new userInterval partition evolutions with emigration related to the Aldous diffusion
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We construct a stationary Markov process corresponding to the evolution of masses and distances of subtrees along the spine from the root to a branch point in a conjectured stationary, continuum random tree-valued diffusion that was proposed by David Aldous. As a corollary this Markov process induces a recurrent extension, with Dirichlet stationary distribution, of a Wright-Fisher diffusion for which zero is an exit boundary of the coordinate processes. This extends previous work of Pal who argued a Wright-Fisher limit for the three-mass process under the conjectured Aldous diffusion until the disappearance of the branch point. In particular, the construction here yields the first stationary, Markovian projection of the conjectured diffusion. Our construction follows from that of a pair of interval partition-valued diffusions that were previously introduced by the current authors as continuum analogues of down-up chains on ordered Chinese restaurants with parameters (1/2,1/2) and (1/2,0). These two diffusions are given by an underlying Crump-Mode-Jagers branching process, respectively with or without immigration. In particular, we adapt the previous construction to build a continuum analogue of a down-up ordered Chinese restaurant process with the unusual parameters (1/2,-1/2), for which the underlying branching process has emigration.
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Forman et al. (2020+) constructed $(alpha,theta)$-interval partition evolutions for $alphain(0,1)$ and $thetage 0$, in which the total sums of interval lengths (total mass) evolve as squared Bessel processes of dimension $2theta$, where $thetage 0$ acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson--Dirichlet interval partitions. In this paper we study symmetry properties of $(alpha,theta)$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${rm SSIP}^{(alpha)}(theta_1,theta_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $theta_1ge 0$ and $theta_2ge 0$. They also have squared Bessel total mass processes of dimension $2theta$, where $theta=theta_1+theta_2-alphage-alpha$ covers emigration as well as immigration. Under the constraint $max{theta_1,theta_2}gealpha$, we prove that an ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $alpha$ and $theta$, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.
The Aldous diffusion is a conjectured Markov process on the space of real trees that is the continuum analogue of discrete Markov chains on binary trees. We construct this conjectured process via a consistent system of stationary evolutions of binary trees with $k$ labeled leaves and edges decorated with diffusions on a space of interval partitions constructed in previous work by the same authors. This pathwise construction allows us to study and compute path properties of the Aldous diffusion including evolutions of projected masses and distances between branch points. A key part of proving the consistency of the projective system is Rogers and Pitmans notion of intertwining.
We introduce a three-parameter family of up-down ordered Chinese restaurant processes ${rm PCRP}^{(alpha)}(theta_1,theta_2)$, $alphain(0,1)$, $theta_1,theta_2ge 0$, generalising the two-parameter family of Rogers and Winkel. Our main result establishes self-similar diffusion limits, ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolutions generalising existing families of interval partition evolutions. We use the scaling limit approach to extend stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $alphain(0,1)$ and $theta:=theta_1+theta_2-alphage-alpha$, including for the first time the usual range of $theta>-alpha$ rather than being restricted to $thetage 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.
Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $Ntimes N$ torus, and the state of the process at time $t$ is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate $N^{-alpha}$ to a site chosen at random from the torus. We will be interested in the case in which $alpha<3$, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically $T=(2-2alpha/3)N^{alpha/3}log N$. If $rho_s$ is the fraction of the population who know the information at time $s$ and $varepsilon$ is small then, for large $N$, the time until $rho_s$ reaches $varepsilon$ is $T(varepsilon)approx T+N^{alpha/3}log (3varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $rho_s$ at time $s=T(1/3)+tN^{alpha/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.
Aldous spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.
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