arXiv 1909.093722019-09-20math-phhep-thmath.MP

回路方程的解是随机矩阵

Solutions of loop equations are random matrices

B. Eynard

备注：: 22 pages + appendix(7pages), Latex, 2 figures

摘要：

对于给定的多项式$V(x)\in \mathbb C[x]$，随机矩阵特征值测度是$γ^N$上的测度$\prod_{1\leq i<j\leq N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}dx_i$。厄米矩阵有实特征值$γ=\mathbb R$，它推广到$γ$一个复杂的约旦弧，或实际上是一个线性组合同伦类约旦弧，选择这样的积分是绝对收敛的。这种测度的多项式矩满足一组称为“循环方程”的线性方程。我们证明了对于某些弧的选择，回路方程的每一个解必然是某个随机矩阵测度的多项式矩。可积弧的同调空间与回路方程的解集之间存在同构。我们也推广到一个2-矩阵模型和矩阵链，并$V$是不是一个多项式，但$V'(x)\in \mathbb C(x)$的情况下。

英文摘要：

For a given polynomial $V(x)\in \mathbb C[x]$, a random matrix eigenvalues measure is a measure $\prod_{1\leq i<j\leq N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}dx_i$ on $γ^N$. Hermitian matrices have real eigenvalues $γ=\mathbb R$, which generalize to $γ$ a complex Jordan arc, or actually a linear combination of homotopy classes of Jordan arcs, chosen such that integrals are absolutely convergent. Polynomial moments of such measure satisfy a set of linear equations called "loop equations". We prove that every solution of loop equations are necessarily polynomial moments of some random matrix measure for some choice of arcs. There is an isomorphism between the homology space of integrable arcs and the set of solutions of loop equations. We also generalize this to a 2-matrix model and to the chain of matrices, and to cases where $V$ is not a polynomial but $V'(x)\in \mathbb C(x)$.

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摘要：

英文摘要：

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