def _compute_joint_eigen_distribution(self, beta):
"""
Helper function to compute the joint distribution of phases
of the complex eigen values of matrices belonging to any
circular ensembles.
"""
n = self.dimension
Zbn = ((2*pi)**n)*(gamma(beta*n/2 + 1)/S(gamma(beta/2 + 1))**n)
t = IndexedBase('t')
i, j, k = (Dummy('i', integer=True), Dummy('j', integer=True),
Dummy('k', integer=True))
syms = ArrayComprehension(t[i], (i, 1, n)).doit()
f = Product(Product(Abs(exp(I*t[k]) - exp(I*t[j]))**beta, (j, k + 1, n)).doit(),
(k, 1, n - 1)).doit()
return Lambda(tuple(syms), f/Zbn)
def _compute_joint_eigen_dsitribution(self, beta):
"""
Helper function for computing the joint
probability distribution of eigen values
of the random matrix.
"""
n = self.dimension
Zbn = self._compute_normalization_constant(beta, n)
l = IndexedBase('l')
i = Dummy('i', integer=True, positive=True)
j = Dummy('j', integer=True, positive=True)
k = Dummy('k', integer=True, positive=True)
term1 = exp((-S(n) / 2) * Sum(l[k]**2, (k, 1, n)).doit())
sub_term = Lambda(i, Product(Abs(l[j] - l[i])**beta, (j, i + 1, n)))
term2 = Product(sub_term(i).doit(), (i, 1, n - 1)).doit()
syms = ArrayComprehension(l[k], (k, 1, n)).doit()
return Lambda(syms, (term1 * term2) / Zbn)