3 research outputs found
Form factor for large quantum graphs: evaluating orbits with time-reversal
It has been shown that for a certain special type of quantum graphs the
random-matrix form factor can be recovered to at least third order in the
scaled time \tau using periodic-orbit theory. Two types of contributing pairs
of orbits were identified, those which require time-reversal symmetry and those
which do not. We present a new technique of dealing with contribution from the
former type of orbits.
The technique allows us to derive the third order term of the expansion for
general graphs. Although the derivation is rather technical, the advantages of
the technique are obvious: it makes the derivation tractable, it identifies
explicitly the orbit configurations which give the correct contribution, it is
more algorithmical and more system-independent, making possible future
applications of the technique to systems other than quantum graphs.Comment: 25 pages, 14 figures, accepted to Waves in Random Media (special
issue on Quantum Graphs and their Applications). Fixed typos, removed an
overly restrictive condition (appendix), shortened introductory section
Spectral statistics for unitary transfer matrices of binary graphs
Quantum graphs have recently been introduced as model systems to study the
spectral statistics of linear wave problems with chaotic classical limits. It
is proposed here to generalise this approach by considering arbitrary, directed
graphs with unitary transfer matrices. An exponentially increasing contribution
to the form factor is identified when performing a diagonal summation over
periodic orbit degeneracy classes. A special class of graphs, so-called binary
graphs, is studied in more detail. For these, the conditions for periodic orbit
pairs to be correlated (including correlations due to the unitarity of the
transfer matrix) can be given explicitly. Using combinatorial techniques it is
possible to perform the summation over correlated periodic orbit pair
contributions to the form factor for some low--dimensional cases. Gradual
convergence towards random matrix results is observed when increasing the
number of vertices of the binary graphs.Comment: 18 pages, 8 figure
Leading off-diagonal correction to the form factor of large graphs
Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(Ï„), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2Ï„2 to the form factor which agrees with random-matrix theory