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  "body": "\n## Statement\n\nLet $(Y_{ij})_{1 \\le i \u003c j}$ be [[independent and identically distributed|00MRN0FARV]] real random variables with mean $0$ and variance $1$.\nLet $(Y_{ii})_{i \\ge 1}$ be independent and identically distributed real random variables with mean $0$ and finite variance, independent of the $Y_{ij}$ with $i \u003c j$.\nFor $n \\ge 1$, let $W_n$ be the real symmetric $n \\times n$ matrix with entries $W_n(i, j) = W_n(j, i) = Y_{ij} / \\sqrt n$ for $i \\le j$.\nLet $\\lambda_1, \\ldots, \\lambda_n$ be the [[eigenvalues|0BQAA2BK3Q]] of $W_n$, and let\n$$\\mu_n = \\frac1n \\sum_{k=1}^n \\delta_{\\lambda_k}$$\nbe its empirical spectral measure.\nThen, almost surely, $\\mu_n$ [[converges weakly|0SRYJGXKR4]] as $n \\to \\infty$ to the semicircle distribution, the probability measure on $\\R$ with density\n$$\\rho(x) = \\frac{1}{2\\pi} \\sqrt{4 - x^2} \\quad \\text{for } \\abs{x} \\le 2, \\qquad \\rho(x) = 0 \\quad \\text{for } \\abs{x} \u003e 2.$$\n\n## Remarks\n\nThe moment method is the classical proof.\nOne shows that $\\mathbb{E} \\int x^k \\, d\\mu_n = \\frac1n \\mathbb{E} \\tr W_n^k$ tends to the $k$-th moment of the semicircle law, which is $0$ for odd $k$ and the Catalan number $\\frac{1}{m+1}\\binom{2m}{m}$ for $k = 2m$.\nExpand the trace as a sum over closed paths.\nIn the limit only paths that trace a tree and cross each edge exactly twice contribute, and Catalan numbers count them.\nA variance bound and a truncation of the entries then give almost sure convergence.\nThe same result holds for complex [[Hermitian|0EKZCX4K3P]] matrices with independent and identically distributed entries above the diagonal, of mean $0$ and $\\mathbb{E}\\abs{Y_{ij}}^2 = 1$.\n\n## Explanations\n\n### L1 human\n\nFill a large symmetric matrix with independent random numbers and scale it by $1/\\sqrt n$.\nIts eigenvalues then spread over the interval $[-2, 2]$, and their histogram takes the shape of a half-disc.\nThe shape does not depend on the law of the entries, only on their mean and variance.\n\n### L2 human\n\nIt is the law of large numbers of random matrix theory, the analogue for spectra of the law of large numbers for averages.\nWigner introduced such matrices as models for the energy levels of heavy atomic nuclei.\nFiner questions, such as the fluctuations of the largest eigenvalue, lead to the [[Tracy–Widom distribution|89ZY5YRETE]].\nFor sample covariance matrices the analogue is the [[Marchenko–Pastur law|DCK63WTY5E]].\n",
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