Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Form factor for a family of quantum graphs: An expansion to third order   For certain types of quantum graphs we show that the random-matrix form
factor can be recovered to at least third order in the scaled time $\tau$ from
periodic-orbit theory. We consider the contributions from pairs of periodic
orbits represented by diagrams with up to two self-intersections connected by
up to four arcs and explain why all other diagrams are expected to give
higher-order corrections only.
  For a large family of graphs with ergodic classical dynamics the diagrams
that exist in the absence of time-reversal symmetry sum to zero. The mechanism
for this cancellation is rather general which suggests that it may also apply
at higher-orders in the expansion. This expectation is in full agreement with
the fact that in this case the linear-$\tau$ contribution, the diagonal
approximation, already reproduces the random-matrix form factor for $\tau<1$.
For systems with time-reversal symmetry there are more diagrams which
contribute at third order. We sum these contributions for quantum graphs with
uniformly hyperbolic dynamics, obtaining $+2\tau^{3}$, in agreement with
random-matrix theory. As in the previous calculation of the leading-order
correction to the diagonal approximation we find that the third order
contribution can be attributed to exceptional orbits representing the
intersection of diagram classes.

35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
2
The categories of this paper are: math
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con