arXiv subjects

Svante Janson

Publications and source records attributed to Svante Janson.

At least 235 records · Page 13Linked to original sources

arXiv2007

Precise logarithmic asymptotics for the right tails of some limit random variables for random trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant's precise value remains unknown.

math.PR↗

arXiv2006

Rounding of continuous random variables and oscillatory asymptotics

We study the characteristic function and moments of the integer-valued random variable $\lfloor X+α\rfloor$, where $X$ is a continuous random variables. The results can be regarded as exact versions of Sheppard's correction. Rounded variables of this type often occur as subsequence limits of sequences of integer-valued random variables. This leads to oscillatory terms in asymptotics for these variables, something that has often been observed, for example in the analysis of several algorithms. We give some examples, including applications to tries, digital search trees and Patricia tries.

math.PR↗

arXiv2006

Spread-out percolation in R^d

Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$ within distance $r$ independently with probability $p$. This is the simplest case of a `spread-out' percolation model studied by Penrose, who showed that, as $r\to\infty$, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of the authors on inhomogeneous random graphs.

math.PR↗

arXiv2006

The probability that a random multigraph is simple

Consider a random multigraph G* with given vertex degrees d_1,...,d_n, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending to infinity, the probability that the multigraph is simple stays away from 0 if and only if \sum d_i^2=O(\sum d_i). This was previously known only under extra assumtions on the maximum degree. We also give an asymptotic formula for this probability, extending previous results by several authors.

math.CO↗

arXiv2006

Complex interpolation of compact operators mapping into the couple (FL^{\infty},FL_{1}^{\infty})

If (A_0,A_1) and (B_0,B_1) are Banach couples and a linear operator T from A_0 + A_1 to B_0 + B_1 maps A_0 compactly into B_0 and maps A_1 boundedly into B_1, does T necessarily also map [A_0,A_1]_s compactly into [B_0,B_1]_s for s in (0,1)? After 42 years this question is still not answered, not even in the case where T is also compact from A_1 to B_1. But affirmative answers are known for many special choices of (A_0,A_1) and (B_0,B_1). Furthermore it is known that it would suffice to resolve this question in the special case where (B_0,B_1) is the special couple (l^\infty(FL^\infty), l^\infty(FL^\infty_1)). Here FL^\infty is the space of all sequences which are Fourier coefficients of bounded functions, FL^\infty_1 is the weighted space of all sequences (a_n) such that (e^n a_n) is in FL^\infty, and thus B_0 and B_1 are the spaces of bounded sequences of elements in these spaces (i.e., they are spaces of doubly indexed sequences). We provide an affirmative answer to this question in the related but simpler case where (B_0,B_1) is the special couple (FL^\infty,FL^\infty_1).

math.FA↗

arXiv2006

The phase transition in inhomogeneous random graphs

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that under one very weak assumption (that the expected number of edges is `what it should be'), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is `stable': for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n).

math.PR↗

arXiv2006

On a random graph related to quantum theory

We show that a random graph studied by Ioffe and Levit is an example of an inhomogeneous random graph of the type studied by Bollobas, Janson and Riordan, which enables us to give a new, simple, proof of their result on a phase transition.

math.PR↗

arXiv2006

The density of the ISE and local limit laws for embedded trees

It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local version of this result: ISE has a (random) Hölder continuous density, and the vertical profile of embedded trees converges to this density, at least for some such trees. As a consequence, we derive a formula for the distribution of the density of ISE at a given point. This follows from earlier results by Bousquet-Mélou on convergence of the vertical profile at a fixed point. We also provide a recurrence relation defining the moments of the (random) moments of ISE.

math.PR↗

arXiv2006

Conditioned Galton-Watson trees do not grow

An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, as has been shown in a special case by Luczak and Winkler.

math.PR↗

arXiv2005

Partial fillup and search time in LC tries

Andersson and Nilsson introduced in 1993 a level-compressed trie (in short: LC trie) in which a full subtree of a node is compressed to a single node of degree being the size of the subtree. Recent experimental results indicated a 'dramatic improvement' when full subtrees are replaced by partially filled subtrees. In this paper, we provide a theoretical justification of these experimental results showing, among others, a rather moderate improvement of the search time over the original LC tries. For such an analysis, we assume that n strings are generated independently by a binary memoryless source with p denoting the probability of emitting a 1. We first prove that the so called alpha-fillup level (i.e., the largest level in a trie with alpha fraction of nodes present at this level) is concentrated on two values with high probability. We give these values explicitly up to O(1), and observe that the value of alpha (strictly between 0 and 1) does not affect the leading term. This result directly yields the typical depth (search time) in the alpha-LC tries with p not equal to 1/2, which turns out to be C loglog n for an explicitly given constant C (depending on p but not on alpha). This should be compared with recently found typical depth in the original LC tries which is C' loglog n for a larger constant C'. The search time in alpha-LC tries is thus smaller but of the same order as in the original LC tries.

cs.DS↗

arXiv2005

Congruence properties of depths in some random trees

Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n tends to infinty. The same is true for the number of vertices of depth divisible by m for m=3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m at least 7, the number is not asymptotically normal, and the variance grows faster than linear in n. The case m=6 is intermediate: the number is asymptotically normal but the variance is of order n log n. This is a simple and striking example of a type of phase transition that has been observed by other authors in several cases. We prove, and perhaps explain, this non-intuitive behavious using a translation to a generalized Polya urn. Similar results hold for a random binary search tree; now the number of vertices of depth divisible by m is asymptotically normal for m at most 8 but not for m at least 9, and the variance grows linearly in the first case both faster in the second. (There is no intermediate case.) In contrast, we show that for conditioned Galton-Watson trees, including random labelled trees and random binary trees, there is no such phase transition: the number is asymptotically normal for every m.

math.PR↗

arXiv2005

A simple solution to the k-core problem

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the k-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer and Wormald on the existence and size of a k-core in G(n,p) and G(n,m). Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.

math.CO↗

arXiv2005

Rainbow Hamilton cycles in random regular graphs

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.

math.CO↗

arXiv2005

A point process describing the component sizes in the critical window of the random graph evolution

We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n,p) in the critical window p=n^{-1}+lambda n^{-4/3}. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small epsilon is almost constant.

math.PR↗

arXiv2005

Robust reconstruction on trees is determined by the second eigenvalue

Consider a Markov chain on an infinite tree T=(V,E) rooted at ρ. In such a chain, once the initial root state σ(ρ) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let μ_j denote the resulting measure for σ(ρ)=j. The resulting measure μ_j is defined on configurations σ=(σ(x))_{x\in V}\in A^V, where A is some finite set. Let μ_j^n denote the restriction of μto the sigma-algebra generated by the variables σ(x), where x is at distance exactly n from ρ. Letting α_n=max_{i,j\in A}d_{TV}(μ_i^n,μ_j^n), where d_{TV} denotes total variation distance, we say that the reconstruction problem is solvable if lim inf_{n\to\infty}α_n>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n\to\infty. In this paper we study the problem of robust reconstruction. Let νbe a nondegenerate distribution on A and ε>0. Let σbe chosen according to μ_j^n and σ' be obtained from σby letting for each node independently, σ(v)=σ'(v) with probability 1-εand σ'(v) be an independent sample from νotherwise. We denote by μ_j^n[ν,ε] the resulting measure on σ'. The measure μ_j^n[ν,ε] is a perturbation of the measure μ_j^n.

math.PR↗

arXiv2005

Individual displacements in hashing with coalesced chains

We study the asymptotic distribution of the displacements in hashing with coalesced chains, for both late-insertion and early-insertion. Asymptotic formulas for means and variances follow. The method uses Poissonization and some stochastic calculus.

math.PR↗

arXiv2004

Branching Processes, and Random-Cluster Measures on Trees

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?

math.PR↗

arXiv2004

Convergence of coined quantum walks on d-dimensional Euclidean space

Coined quantum walks may be interpreted as the motion in position space of a quantum particle with a spin degree of freedom; the dynamics are determined by iterating a unitary transformation which is the product of a spin transformation and a translation conditional on the spin state. Coined quantum walks on the d-dimensional lattice can be treated as special cases of coined quantum walks on d-dimensional Euclidean space. We study quantum walks on d-dimensional Euclidean space and prove that the sequence of rescaled probability distributions in position space associated to the unitary evolution of the particle converges to a limit distribution.

quant-ph↗