UPDATE: of this file: 12/2/2023

The files all*pdf contain additional figures with respect to 

Nonlinear perturbation of Random Matrix Theory
by K. M. Frahm and D. L. Shepelyansky
Laboratoire de Physique Theorique, Universite de
Toulouse, CNRS, UPS, 31062 Toulouse, France

arXiv:2212.11955
https://www.quantware.ups-tlse.fr/QWLIB/nonlinrmt/index.html


Each pdf file contains several figures or larger size 
(i.e. higher resolution as figures in paper/SupMat). Typically 
the figures are quite self-explaining with labels (for axis, data points 
and additional labels).
All figures here apply to the same RMT, DANSE, realization etc.
(if same model and same N).


1) allSE.pdf => 
both theoretical S(E) curves + numerical data S(E) data for GOE, N=64, 
many beta values 
and for times t=2^15, 2^18, 2^21, 2^24 indicating 
that the numerical values of $\rho_m$ have been obtained 
from the average $<|C_m(t')|^2>$ for $t/2 <= t' <= t$ 
and showing the speed of thermalization for cases where 
thermalization is present (typically slower for smaller/modest beta 
and faster for larger beta).
For each case there are two versions:
(i) S(E) versus E_{m_0} with E_{m_0} being the energy of the initial state
(ii) S(E) versus <E>=\sum_m E_m \rho_m being the average (linear) energy 
of the data which gives (for larger/medium beta) a horizontal shift such that 
typically data points are closer on the EQ-curve. For beta=1.5, 2 at 
E close to +1 there are outside points for (i) which go inside for (ii) 
due to a quite large effect of the non-linear term on the energy. 
For intermediate \beta (e.g. 0.25, 0.35) there is a mix of presence/absence 
of thermalization depending on E_{m_0} due to effects of a chaos border 
depending on E_{m_0} in a complicated way for the given RMT matrix 
realization. 
These figures are generalizations of Fig. 1.


2) all_init_32_SE_states.pdf
all_init_64_SE_states.pdf
all_init_128_SE_states.pdf
=> pdf files for GOE, beta=1, 3 values of N=32, 64, 128 with two S(E) 
curves (of type (i) and (ii)) for 4 time values (may be different from 1)
and a full set of state files for all $m_0=1,...,N$ 
showing rho_m versus E_m for two times t with time 
average interval [t/2, t] for rho_m. Furthermore, there are 
the two theoretical curves for EQ and BE using respectif T and 
\mu values using the average linear energy <E>.
These figure are generalizations of Fig. 3  and Fig. S4 and there 
are also related to Figs. 2, S2, S3 (for S(E) or mu, T dependence on E).
Update: Further files for N=256:
all initial m_0 with t=2^22:
----------------------------
updated files 15 Feb 2023 for N=256
=>
all_init_256_1_2E22_SE_states.pdf (beta=1)
all_init_256_2_2E22_SE_states.pdf (beta=2)
first 35 iniital m_0 with t=2^27 and two RMT realisations):
-----------------------------------------------------------
all_init_256_1_Real1_2E27_SE_states.pdf (1st realisation)
all_init_256_1_Real2_2E27_SE_states.pdf (2nd realisation)


3) all_initAnderson1d2_64_SE_states.pdf
all_initAnderson1d4_64_SE_states.pdf
=> same as 2) but for the DANSE model with W=2,4 and N=64, 
generalizations of Fig. S12 and related to Fig. S11.
Here thermalization is possible but with stronger fluctuations as for GOE, 
and more for W=2 than for W=4 and longer time scales. 


4) all_F0.5_64_SE_states.pdf
all_F0.5_32_SE_states.pdf
all_F0.25_64_SE_states.pdf
all_F0.25_32_SE_states.pdf
=> same as 2) but for the GOE with additional diagonal matrix 
elements fn for N=32,N=64, f=F=0.25,0.5 and \beta=2
generalizations of Figs. S13 and S14.
Here thermalization is very slow if present (f=0.25 or N=32)
or even mostly absent (f=0.5 and N=64). This corresponds physically 
roughly to the GPE case (Sinai ocscillator and Bunimovich stadium, 
refs. [32-34]).


5) alldist.pdf
=> Distribution p(x) and integrated distribution 
P(x)=\int_x^\infty p(y) dy 
of the rescaled variable x=(E-\mu)|C_m|^2/T with T and \mu determined 
from the EQ ansatz (parameters: N=64, beta=1, GOE case).
Theoretically both distributions are supposed to be: p(x)=P(x)=e^{-x}.
There are 2x4x4 cases (8 figures) for p(x),P(x), 
m_0=9,17,25,33 and m=9,17,25,33.
For P(x) and certain p(x) cases one can see a saturation effect 
with p(x) or P(x) below the theoretical curve for larger x>8-10 
which is due to finite N such that the formula:

(1-x/N)^N \approx e^{-x} 

is not exactly valid (the theoretical justification of the 
gaussian/exponential marginal distribution for the micro-canonical 
ensemble is based and that formula). 
Apart from this point, for "smaller/modest" x<6-8 and despite 
the quite small bin-value of 0.05 for histograms (and 0.005 for P(x)) 
the numerical data 
coincides very accurately with the theoretical distribution. 
The histograms have been computed with a lot of data points 10*2^23 
(factor 10 for each \Delta t=0.1 step). However, the statistical 
quality of the histograms correspond to a "smaller" number of data 
points due to rather long time correlations (\sim 10^2-10^3 depending 
on m_0 and m) for the time dependence of $|C_m(t)|^2$.

updates 15 Feb 2023 for Lyapunove exponent below

6) all_lyap_Nscale.pdf
N-scaling of Lyapunov exponent for beta=1 with N up to N=512 
for iniitial energies close to zero for different quantities (average,
all data, fixed ratio m_0/N; localized initial conditions).
The green power law fit corresponds to data for N<= 128 and 
the pink power law fit corresponds to data for N>= 192.
The blue line gives a 1/N^2 fit with finite size correction.
Generalisation of Fig. S8.


7) all_lyap_beta_scale_random_init.pdf
beta-scaling of Lyapunov exponent for N=64 and 32, for 0.02<=beta<=2 
and random uniform initial conditions
Two types of fit (power law and a\beta+b\beta^2) for 
average and all data.
The first two pages show the energy dependence of \lambda 
(concentrated around zero due to uniform random initial conditions).
Generalisation of bottom panel of Fig. S7 (and of Fig. S6 for first 
two pages).


8) all_lyap_long_time_local_init.pdf
time evolution of Lyapunov exponent for N=32, 64 
and certain beta values for the case of localized initial conditions.
At smallest beta=0.02 most values continue to decay indicating absence 
of chaos but a few values stay stable.
Generalisation of Fig. S10.

9) all_lyap_long_time_local_init.pdf
as 8) for 35 random initial conditions, beta=0.02, N=32, 64 and 
very long times.
Generalisation of Fig. S10.