arXiv subjects
Publications and source records attributed to David Aldous.
Consider routing traffic on the N x N torus, simultaneously between all source-destination pairs, to minimize the cost $\sum_ec(e)f^2(e)$, where f(e) is the volume of flow across edge e and the c(e) form an i.i.d. random environment. We prove existence of a rescaled $N\to \infty$ limit constant for minimum cost, by comparison with an appropriate analogous problem about minimum-cost flows across a M x M subsquare of the lattice.
We study the relation between the minimal spanning tree (MST) on many random points and the "near-minimal" tree which is optimal subject to the constraint that a proportion $δ$ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as $1 + Θ(δ^2)$. We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.
Consider networks on $n$ vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as $n^{13/10}$. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order $n^{3/2}$. Finally, if we require the network length to be O(n) then the mean number of hops grows as order $\log \log n$. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense.
We prove consistency of four different approaches to formalizing the idea of minimum average edge-length in a path linking some infinite subset of points of a Poisson process. The approaches are (i) shortest path from origin through some $m$ distinct points; (ii) shortest average edge-length in paths across the diagonal of a large cube; (iii) shortest path through some specified proportion $δ$ of points in a large cube; (iv) translation-invariant measures on paths in $\Reals^d$ which contain a proportion $δ$ of the Poisson points. We develop basic properties of a normalized average length function $c(δ)$ and pose challenging open problem
In a network where the cost of flow across an edge is nonlinear in the volume of flow, and where sources and destinations are uniform, one can consider the relationship between total volume $v$ of flow through the network and the minimum cost $c = Psi(v)$ of any flow with volume $v$. Under a simple probability model (locally tree-like directed network, independent cost-volume functions or different edges) we show how to compute $Ψ(v)$ in the infinite-size limit. The argument uses a probabilistic reformulation of the cavity method from statistical physics, and is not rigorous as presented here. The methodology seems potentially useful for many problems concerning flows on this class of random networks.
Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1.
We consider combinatorial optimization problems defined over random ensembles, and study how solution cost increases when the optimal solution undergoes a small perturbation delta. For the minimum spanning tree, the increase in cost scales as delta^2; for the mean-field and Euclidean minimum matching and traveling salesman problems in dimension d>=2, the increase scales as delta^3; this is observed in Monte Carlo simulations in d=2,3,4 and in theoretical analysis of a mean-field model. We speculate that the scaling exponent could serve to classify combinatorial optimization problems into a small number of distinct categories, similar to universality classes in statistical physics.
We study a reversible one-dimensional spin system with Bernoulli(p) stationary distribution, in which a site can flip only if the site to its left is in state +1. Such models have been used as simple exemplars of systems exhibiting slow relaxation. We give fairly sharp estimates of the spectral gap as p decreases to zero. The method uses Poincare comparison with a long-range process which is analyzed by probabilistic methods (coupling, supermartingales).