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10
rank
1
0.01
0.1
l
Fig. 6.4. Rank{to{eigenvalue plot for the discrete Laplacian matrix of the 0-temperature connec-
tivity graph for the homopolymer (empty circles), the slow-folding heteropolymer (lled squares)
and the fast-folding heteropolymer (crosses). The three dashed lines refer to power{law least
square ts of exponents 3.2, 3.3 and 3.4 respectively. In order to ease the distinction between
dierent curves the x-variable of the curve referring to the slow-folder has been multiplied by an
arbitrary factor.
t of the low frequency part of the spectral density. This last quantity approximates
the spectral dimension in the sense that it converges to it as the graph size goes to
innity. It still however retain the same basic meaning also for nite system sizes.
In fact, while in nite graphs, as those under study, the recurrence and rst passage
times are always nite, a lower d still implies a higher diculty in exploring the
graph. In order to better elucidate this concept we recall that the spectral dimen-
sion is computed by diagonalizing the discrete Laplacian matrix, that is a Laplacian
matrix where o{diagonal entries do not correspond to the full transition rates but
only to zeroes and ones depending on the existence of a connection between the
corresponding nodes. Such a matrix retains the information about the graph topol-
ogy but looses any information about the kinetics of the system. As a consequence
its spectral properties have to be interpreted in terms of a discrete hopping process
between nodes: the eigenvalues of the discrete Laplacian matrix don't relate to time
scales but rather to path lengths. In this respect high spectral dimensions imply a
higher density of long relaxation paths.
In Fig. 6.4 we plot the rank of the rst 30 eigenvaluess of the discrete Laplacian
matrix against their numerical value for each of the zero{temperature connectivity
graphs associated with the three analyzed sequences. The resulting curves are
proportional to the integral of the spectral density of the discrete Laplacian matrix,
and can be thus least{square{tted with a power{law in order to estimate the
corresponding spectral dimension, which is equal to twice the tted exponent. This
procedure holds an essentially constant result, namely d = 6:4 for the homopolymer,
d = 6:5 for the slow{folding heteropolymer and d = 6:8 for the fast{folding one.
