279 research outputs found
Statistics on Graphs, Exponential Formula and Combinatorial Physics
The concern of this paper is a famous combinatorial formula known under the
name "exponential formula". It occurs quite naturally in many contexts
(physics, mathematics, computer science). Roughly speaking, it expresses that
the exponential generating function of a whole structure is equal to the
exponential of those of connected substructures. Keeping this descriptive
statement as a guideline, we develop a general framework to handle many
different situations in which the exponential formula can be applied
Constructive Field Theory in Zero Dimension
In this pedagogical note we propose to wander through five different methods
to compute the number of connected graphs of the zero-dimensional
field theory,in increasing order of sophistication. The note does not contain
any new result but may be helpful to summarize the heart of constructive
resummations, namely a replica trick and a forest formula.Comment: 12 pages,one figur
Services within a busy period of an M/M/1 queue and Dyck paths
We analyze the service times of customers in a stable M/M/1 queue in
equilibrium depending on their position in a busy period. We give the law of
the service of a customer at the beginning, at the end, or in the middle of the
busy period. It enables as a by-product to prove that the process of instants
of beginning of services is not Poisson. We then proceed to a more precise
analysis. We consider a family of polynomial generating series associated with
Dyck paths of length 2n and we show that they provide the correlation function
of the successive services in a busy period with (n+1) customers
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