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rank
1
0.01
0.1
l
Fig. 6.5. Rank{to{eigenvalue plot for the discrete Laplacian matrix of the homopolymer for
ve dierent temperatures: T = 0:00; 0:02; 0:04; 0:06; 0:08. Increasing circle size corresponds to
increasing temperature. Continuous lines refer to power{law least{square ts of exponents 3.2,
3.1, 3.05 and 3.1 respectively. In order to ease the distinction of dierent curves the x-variable of
each curve has been multiplied by an arbitrary factor.
It is important to notice that the spectral dimension is a very robust quantity,
being invariant for topological rescaling such as the renormalizing procedure we
introduced, local link redirection, decimation or creation. This property, which is
again rigorously true only in case of innite graphs, still holds approximately for
suciently large graphs. Such topological invariance is very useful when dealing
with graphs that are a sample of other graphs, because it implies that the spectral
dimension will not change dramatically if some connections are lost or wrongly
assigned.
In order to study how d behaves with temperature we repeat the same eval-
uation procedure for the renormalized graph at the four usual temperatures:
T = 0:02; 0:04; 0:06; 0:08. The resulting rank{to{eigenvalue plots are shown, to-
gether with the 0-temperature case, in Figs. 6.5, 6.6, 6.7. While Fig. 6.5 shows
that, in the case of the homopolymer, the measured spectral dimension does not
change with temperature, the two gures referring to the heteropolymers (6.7, 6.6)
show a substantial decrease in this quantity for increasing temperatures. More pre-
cisely, while the spectral dimension of the zero{temperature connectivity graph of
the two heteropolymers is around 6:5, at higher temperatures it stabilizes around
5, suggesting that the conguration space of heteropolymers appears more compact
to a random exploration than that of a homopolymer.
In order to rule out that this change in the dimensionality of the system is a nite
size eect we recall that between T = 0:02 and T = 0:08, a temperature range where
spectral dimension is constant for all sequences, renormalization causes the number
of nodes of the connectivity graph of the heteropolymers to decrease by an order
of magnitude (see Fig. 6.1). Also the graph diameter, measured as the maximum
