The volumes of Kostka polytopes appear naturally in questions of random matrix theory in the context of the randomized Schur-Horn problem, i.e., evaluating the probability density that a random Hermitian matrix with fixed spectrum has a given diagonal. We give a polynomial-time deterministic algorithm for approximating the volume of a ($\Omega(n^2)$ dimensional) Kostka polytope $\mathrm{GT}(\lambda, \mu)$ to within a multiplicative factor of $\exp(O(n\log n))$, when $\lambda$ is an integral partition with $n$ parts, with entries bounded above by a polynomial in $n$, and $\mu$ is an integer vector lying in the interior of the Schur-Horn polytope associated to $\lambda$. The algorithm thus gives asymptotically correct estimates of the log-volume of Kostka polytopes corresponding to such $(\lambda, \mu)$. Our approach is based on a partition function interpretation of the continuous analogue of Schur polynomials, and an associated maximum entropy principle.

翻译：Kostka多面体体积问题随机出现在随机矩阵理论中随机Schur-Horn问题的研究背景下，即评估具有固定谱的随机埃尔米特矩阵具有给定对角元的概率密度。当$\lambda$是包含$n$个分量的整数分割且其分量值以$n$的多项式为上界，同时$\mu$是位于与$\lambda$相关的Schur-Horn多面体内部的整数向量时，我们提出了一种多项式时间确定性算法，用于在$\exp(O(n\log n))$乘法因子内逼近($\Omega(n^2)$维)Kostka多面体$\mathrm{GT}(\lambda, \mu)$的体积。该算法从而为对应此类$(\lambda, \mu)$的Kostka多面体的对数体积提供了渐近正确的估计。我们的方法基于Schur多项式连续模拟的配分函数解释，以及相关的最大熵原理。