Random Matrices

Circular Ensembles (COE, CUE, ...)

The circular ensembles are families of unitary matrices with distributions invariant under different unitary transformations. Typical applications are in statistical mechanics, number theory, combinatorics, and nuclear physics.

Matrices from the circular real ensemble (CRE) are orthogonal. »

In[1]:=
Click for copyable input
cre = RandomVariate[CircularRealMatrixDistribution[5]];
In[2]:=
Click for copyable input
OrthogonalMatrixQ[cre]
Out[2]=

Matrices from the circular unitary ensemble (CUE) are unitary. »

In[3]:=
Click for copyable input
cue = RandomVariate[CircularUnitaryMatrixDistribution[5]];
In[4]:=
Click for copyable input
UnitaryMatrixQ[cue]
Out[4]=

Matrices from the circular orthogonal ensemble (COE) are symmetric and unitary. »

In[5]:=
Click for copyable input
coe = RandomVariate[CircularOrthogonalMatrixDistribution[5]];
In[6]:=
Click for copyable input
SymmetricMatrixQ[coe] && UnitaryMatrixQ[coe]
Out[6]=

Matrices from the circular symplectic ensemble (CSE) are self-dual unitary quaternionic. »

show complete Wolfram Language input
In[7]:=
Click for copyable input
selfdualQuaternionicQ[m_] := With[{\[ScriptCapitalJ] = KroneckerProduct[{{0, -1}, {1, 0}}, IdentityMatrix[Length[m]/2]], mat = SetAccuracy[m, 10]}, Transpose[mat].\[ScriptCapitalJ] == \[ScriptCapitalJ].mat];
In[8]:=
Click for copyable input
cse = RandomVariate[CircularSymplecticMatrixDistribution[5]];
In[9]:=
Click for copyable input
UnitaryMatrixQ[cse] && selfdualQuaternionicQ[cse]
Out[9]=

Matrices from the circular quaternion ensemble (CQE) are symplectic unitary. »

show complete Wolfram Language input
In[10]:=
Click for copyable input
symplecticMatrixQ[mat_] := With[{\[ScriptCapitalJ] = KroneckerProduct[{{0, -1}, {1, 0}}, IdentityMatrix[Length[mat]/2]] }, Conjugate[mat].\[ScriptCapitalJ] == \[ScriptCapitalJ].mat];
In[11]:=
Click for copyable input
cqe = RandomVariate[CircularQuaternionMatrixDistribution[5]];
In[12]:=
Click for copyable input
UnitaryMatrixQ[cqe] && symplecticMatrixQ[cqe]
Out[12]=

Eigenvalues of matrices from CUE, COE, and CSE have unit length and are uniformly distributed in phase.

show complete Wolfram Language input
In[13]:=
Click for copyable input
args = Flatten[ Arg[RandomVariate[ MatrixPropertyDistribution[Eigenvalues[x], x \[Distributed] #], 10^4]]] & /@ {CircularUnitaryMatrixDistribution[5], CircularOrthogonalMatrixDistribution[5], CircularSymplecticMatrixDistribution[5]}; Row[MapThread[ Histogram[#1, {-Pi, Pi, Pi/10}, Frame -> None, ChartLegends -> Placed[#2, Above]] &, {args, {Style["Unitary", 15], Style["Orthogonal", 15], Style["Symplectic", 15]}}]]
Out[13]=

Visualize the joint distribution of the phase of the eigenvalues from two-dimensional CUE and compare it with the actual density.

In[14]:=
Click for copyable input
evs\[ScriptCapitalD] = MatrixPropertyDistribution[Arg[Eigenvalues[x]], x \[Distributed] CircularUnitaryMatrixDistribution[2]]; \[CurlyPhi]s = RandomSample /@ RandomVariate[evs\[ScriptCapitalD], 10^5];
show complete Wolfram Language input
In[14]:=
Click for copyable input
Show[ Histogram3D[\[CurlyPhi]s, {-Pi, Pi, 0.25}, PDF, PlotTheme -> "Scientific", ChartStyle -> "AvocadoColors"], Plot3D[1/(8 Pi^2) Abs[Exp[I \[Phi]1] - Exp[I \[Phi]2]]^2, {\[Phi]1, -Pi, Pi}, {\[Phi]2, -Pi, Pi}, PlotStyle -> None, MeshStyle -> Thick], ImageSize -> Medium]
Out[15]=

Related Examples

de es fr ja ko pt-br ru zh