Svante Janson

Gaussian Hilbert spaces

1. Gaussian Hilbert spaces 2. Wiener chaos 3. Wick products 4. Tensor products and Fock spaces 5. Hypercontractivity 6. Distributions of variables with finite chaos expansions 7. Stochastic integration 8. Gaussian stochastic processes 9. Conditioning 10. Limit theorems for generalized U-statistics 11. Applications to operator theory 12. Some operators from quantum physics 13. The Cameron-Martin shift 14. Malliavin calculus 15. Transforms Appendices.

Measures of Ecological Association

SummarySix criteria suitable for measures of ecological coexistence are proposed. For twenty such measures are examined whether they satisfy these criteria or not. Four of them satisfy all six criteria. Three of them, suggested by Ochiai, Dice and Jaccard are recommended. For them asymptotic standard errors are given. An example is given with asymptotic confidence intervals for the three measures recommended.

Weak limits for quantum random walks

We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With X(n) denoting position at time n, we show that X(n)/n converges weakly as n--> infinity to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods.

The probability that a random multigraph is simple

Consider a random multigraph G* with given vertex degrees d1,…,dn, constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges , the probability that the multigraph is simple stays away from 0 if and only if . This was previously known only under extra assumptions on the maximum degree maxidi. We also give an asymptotic formula for this probability, extending previous results by several authors.

Minimal and maximal methods of interpolation

Abstract We show that several interpolation functors, including the widely used real and complex methods, are minimal or maximal extensions from a single couple of Banach spaces. Various consequences are drawn from this property.

Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation

We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model. This survey paper contains many known results from many different sources, together with some new results.

Random regular graphs: Asymptotic distributions and contiguity

The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete Levy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r > 3, then the usual (uniformly distributed) random r -regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed.

One, Two and Three Times log n / n for Paths in a Complete Graph with Random Weights

Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n/n for two given points, that the maximum if one point is fixed and the other varies is 2 log n/n, and that the maximum over all pairs of points is 3 log n/n.Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths.

Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wrights constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.

UPPER TAILS FOR SUBGRAPH COUNTS IN RANDOM GRAPHS

LetG be a fixed graph and letXG be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofXG which are best possible up to a logarithmic factor in the exponent. Our argument relies on an extension of Alon’s result about the maximum number of copies ofG in a graph with a given number of edges. Similar bounds are proved for the random graphG(n, M) too.