4,232 research outputs found

### Exchangeable pairs, switchings, and random regular graphs

We consider the distribution of cycle counts in a random regular graph, which
is closely linked to the graph's spectral properties. We broaden the asymptotic
regime in which the cycle counts are known to be approximately Poisson, and we
give an explicit bound in total variation distance for the approximation. Using
this result, we calculate limiting distributions of linear eigenvalue
functionals for random regular graphs.
Previous results on the distribution of cycle counts by McKay, Wormald, and
Wysocka (2004) used the method of switchings, a combinatorial technique for
asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and
demonstrates an interesting connection between the two techniques.Comment: Very minor changes; 23 page

### Cycles and eigenvalues of sequentially growing random regular graphs

Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels
along with their transposes. The resulting matrix is the adjacency matrix of a
random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that
the distribution of smooth linear eigenvalue statistics of this matrix is given
asymptotically by sums of Poisson random variables. This is in contrast with
Gaussian fluctuation of similar quantities in the case of Wigner matrices. It
is also known that for Wigner matrices the joint fluctuation of linear
eigenvalue statistics across minors of growing sizes can be expressed in terms
of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic
(in $n$) fluctuation for a coupling of all random regular graphs of various
degrees obtained by growing each component permutation according to the Chinese
Restaurant Process. Our primary result is that the corresponding eigenvalue
statistics can be expressed in terms of a family of independent Yule processes
with immigration. These processes track the evolution of short cycles in the
graph. If we now take $d$ to infinity, certain GFF-like properties emerge.Comment: Published in at http://dx.doi.org/10.1214/13-AOP864 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Quantitative Small Subgraph Conditioning

We revisit the method of small subgraph conditioning, used to establish that
random regular graphs are Hamiltonian a.a.s. We refine this method using new
technical machinery for random $d$-regular graphs on $n$ vertices that hold not
just asymptotically, but for any values of $d$ and $n$. This lets us estimate
how quickly the probability of containing a Hamiltonian cycle converges to 1,
and it produces quantitative contiguity results between different models of
random regular graphs. These results hold with $d$ held fixed or growing to
infinity with $n$. As additional applications, we establish the distributional
convergence of the number of Hamiltonian cycles when $d$ grows slowly to
infinity, and we prove that the number of Hamiltonian cycles can be
approximately computed from the graph's eigenvalues for almost all regular
graphs.Comment: 59 pages, 5 figures; minor changes for clarit

### On Universal Cycles for Multisets

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of
$\binom{n+t-1}{t}$ integers from [n] with the property that each t-multiset of
[n] appears exactly once consecutively in the sequence. For such a sequence to
exist it is necessary that n divides $\binom{n+t-1}{t}$, and it is reasonable
to conjecture that this condition is sufficient for large enough n in terms of
t. We prove the conjecture completely for t in {2,3} and partially for t in
{4,6}. These results also support a positive answer to a question of Knuth.Comment: 14 pages, two figures, will appear in Discrete Mathematics' special
issue on de Bruijn Cycles, Gray Codes and their generalizations; paper
revised according to journal referees' suggestion

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