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Generating Random Matrices with Normalized Spectral Ranges (Girko Circular Law)
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2021-08-29_mg_random_matrices_circular_law.ipynb
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2021-08-30_mg_random_matrices_circular_law_jl.ipynb
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"source": [
"## Generating Random Matrices with Normalized Spectral Ranges in Julia"
]
},
{
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"source": [
"using Plots\n",
"using Random\n",
"using Distributions\n",
"using LinearAlgebra\n",
"using SparseArrays"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"ExecuteTime": {
"end_time": "2021-08-29T14:31:32.323000-04:00",
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{
"data": {
"text/plain": [
"random_complex_matrix (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function random_complex_matrix(N, ρ)\n",
" σ = 1/√N\n",
" d = Normal(0.0, σ)\n",
" H = ρ * (rand(d, (N, N)) + rand(d, (N, N)) * 1im) / √2\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 3,
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"outputs": [
{
"data": {
"text/plain": [
"random_real_matrix (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function random_real_matrix(N, ρ)\n",
" σ = 1/√N\n",
" d = Normal(0.0, σ)\n",
" H = ρ * rand(d, (N, N))\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
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"end_time": "2021-08-29T14:31:32.608000-04:00",
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"outputs": [
{
"data": {
"text/plain": [
"random_hermitian_matrix (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function random_hermitian_matrix(N, ρ)\n",
" σ = 1/√N\n",
" d = Normal(0.0, σ)\n",
" X = rand(d, (N, N))\n",
" H = ρ * (X + X') / (2*√2)\n",
"end"
]
},
{
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"outputs": [
{
"data": {
"text/plain": [
"plot_evals (generic function with 2 methods)"
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"execution_count": 5,
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"source": [
"function plot_evals(H)\n",
" evals = eigvals(H)\n",
" X = real(evals)\n",
" Y = imag(evals)\n",
" if maximum(Y) > 1e-12\n",
" scatter(X, Y, aspect_ratio=:equal)\n",
" else\n",
" scatter(X, Y, ylim=(-0.1, 0.1))\n",
" end\n",
"end\n",
"\n",
"plot_evals(H::SparseArrays.AbstractSparseMatrix) = plot_evals(Array(H))"
]
},
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},
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}
],
"source": [
"plot_evals(random_complex_matrix(1000, 10))"
]
},
{
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"metadata": {
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"end_time": "2021-08-29T14:31:54.388000-04:00",
"start_time": "2021-08-29T18:31:10.843Z"
}
},
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],
"source": [
"plot_evals(random_real_matrix(1000, 10))"
]
},
{
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"end_time": "2021-08-29T14:31:54.998000-04:00",
"start_time": "2021-08-29T18:31:10.850Z"
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],
"source": [
"plot_evals(random_hermitian_matrix(1000, 10))"
]
},
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"data": {
"text/plain": [
"random_complex_sparse_matrix (generic function with 1 method)"
]
},
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],
"source": [
"function random_complex_sparse_matrix(N, ρ, density)\n",
" σ = 1/√(density * N)\n",
" d = Normal(0.0, σ)\n",
" Hre = sprand(N, N, density, (dims...) -> rand(d, dims...))\n",
" Him = sprand(N, N, density, (dims...) -> rand(d, dims...))\n",
" H = ρ * (Hre + Him * 1im) / √2\n",
"end"
]
},
{
"cell_type": "code",
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"outputs": [
{
"data": {
"text/plain": [
"random_real_sparse_matrix (generic function with 1 method)"
]
},
"execution_count": 10,
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}
],
"source": [
"function random_real_sparse_matrix(N, ρ, density)\n",
" σ = 1/√(density * N)\n",
" d = Normal(0.0, σ)\n",
" H = ρ * sprand(N, N, density, (dims...) -> rand(d, dims...))\n",
"end"
]
},
{
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"execution_count": 11,
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"outputs": [
{
"data": {
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"random_hermitian_sparse_matrix (generic function with 1 method)"
]
},
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],
"source": [
"function random_hermitian_sparse_matrix(N, ρ, density)\n",
" σ = 1/√(density * N)\n",
" d = Normal(0.0, σ)\n",
" H = sprand(N, N, density, (dims...) -> rand(d, dims...))\n",
" return 0.5ρ * (H + H') / √2\n",
"end"
]
},
{
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Raw
Manifest.toml
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Raw
Project.toml
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