% Yi-Kai Liu
% 5/9/01

% Plot the distribution of eigenvalues, and 
% the distribution of nearest-neighbor spacings. 

% FIXME: Look at spacings between eigenvalues in middle of range *only*
% (so that density of eigenvalues in the sample is roughly constant). 
% Another way is to change variables to compensate for the expected 
% semicircular distribution, thus forcing the density to stay constant. 

% OK: Figure out scaling, and plot expected curves

% OK: Vertical scaling is OK. The area under the curves appears 
% to be less than 1, but this is because of the histogram scaling. 


% Requires the following variables: 
%     distribution_name, save_to_disk, base_filename, 
%     N, num_matrices, d, sp

dnumbins = 60;   % Number of bins to plot eigenvalues
spnumbins = 120;  % Number of bins to plot spacings


% Plot distribution of eigenvalues, and the semicircle

[ddist, dbins] = hist(d(:), dnumbins);

figure
bar(dbins, ddist ./ (num_matrices*N))
title(['Distribution of eigenvalues--', distribution_name, ', ',... 
      'N=', num2str(N), ', ', num2str(num_matrices), ' matrices']);

hold on
t = -1.0 : .02 : 1.0;
dbinwidth=2.0/dnumbins;
plot(t, dbinwidth*(2.0/pi)*sqrt(1-t.^2), 'r');  % Semicircle distribution
hold off

if save_to_disk == 1
   saveas(gcf, [base_filename, '-a.fig']);
end;


% Plot distribution of spacings, and the Wigner surmise

[spdist, spbins] = hist(sp(:), spnumbins);

figure
bar(spbins, spdist ./ (num_matrices*(N-1)));
title(['Distribution of spacings--', distribution_name, ', ',... 
      'N=', num2str(N), ', ', num2str(num_matrices), ' matrices'])

hold on
s = 0.0 : .02 : spbins(spnumbins);
spbinwidth = spbins(spnumbins)/spnumbins;
plot(s, spbinwidth*(pi*s/2.0).*exp(-pi*(s.^2)/4.0), 'r');  % Wigner surmise
hold off

if save_to_disk == 1
   saveas(gcf, [base_filename, '-b.fig']);
end;


% Plot distribution of spacings, and the Wigner surmise. 
% Use only spacings from the middle 3/5 of the semicircle 
% (where the eigenvalue density is roughly constant). 

% Pick the spacings between the middle 3/5 of the eigenvalues, 
% then renormalize the spacings to have mean=1. 
sp_midpart = sp(1+round(0.2*(N-2)):1+round(0.8*(N-2)), :);
for m = 1 : num_matrices
   sp_midpart(:,m) = sp_midpart(:,m) ./ mean(sp_midpart(:,m));
end

spnumbins = round(spnumbins*0.6);
[spdist, spbins] = hist(sp_midpart(:), spnumbins);

figure
bar(spbins, spdist ./ (num_matrices*(1+0.6*(N-2))));
title(['Distribution of spacings (from middle 3/5 of semicircle)--',... 
      distribution_name, ', ',... 
      'N=', num2str(N), ', ', num2str(num_matrices), ' matrices']);

hold on
s = 0.0 : .02 : spbins(spnumbins);
spbinwidth = spbins(spnumbins)/spnumbins;
plot(s, spbinwidth*(pi*s/2.0).*exp(-pi*(s.^2)/4.0), 'r');  % Wigner surmise
hold off

if save_to_disk == 1
   saveas(gcf, [base_filename, '-c.fig']);
end;