泊松列联表、吉布斯排列与熵最优传输
Poisson contingency tables, Gibbs permutations, and entropic optimal transport
AI总结：
该研究通过将泊松列联表与Mallows型吉布斯排列耦合，在有限维及渐近尺度上分析其性质，证明了Pal和Mukherjee的相关猜想，涉及熵最优传输等理论。
摘要：
我们研究通过将独立泊松项构成的矩阵按其行和与列和进行条件化得到的随机列联表。我们将此类表与从Mallows型吉布斯测度中抽取的随机排列的块计数对应起来，这是一种精确耦合，其归一化常数可还原Barvinok的永久恒等式。我们利用这种对应关系在三个尺度上研究该表：对于有限维情形，一种转移原理将该表与均值构成静态薛定谔桥（离散熵最优传输问题的解）的独立泊松项矩阵进行比较；随着维度增长、边际和先验均值收敛，该表的归一化直方图以定量速率收敛到极限薛定谔桥。该表围绕桥的波动以两种互补方式被刻画：线性统计量对有界检验函数满足中心极限定理，其极限方差等于在桥下检验函数到可加可分函数子空间的L²距离的平方；作为过程，它们收敛到强度由桥给定、且条件为边际消失的白噪声。从谱角度看，中心化表的极限谱分布由戴森方程支配，其方差轮廓为桥的密度。关键要素是吉布斯测度配分函数的渐近公式，该公式由桥的典范相关的弗雷德霍姆行列式给出。对于Mallows型随机排列，我们的结果在更广泛的一般性下证明了关于极限配分函数的Pal猜想和关于线性统计量中心极限定理的Mukherjee猜想。
英文摘要：
We study random contingency tables obtained by conditioning a matrix of independent Poisson entries on its row and column sums. We identify such a table with the block counts of a random permutation drawn from a Gibbs measure of Mallows type, an exact coupling whose normalizing constants recover Barvinok's permanent identity. We use this correspondence to study the table at three scales. For finite dimensions, a transference principle compares the table with a matrix of independent Poisson entries whose means form the static Schrödinger bridge, the solution of a discrete entropic optimal transport problem. As the dimensions grow and the margins and prior means converge, the normalized histogram of the table converges to a limiting Schrödinger bridge at a quantitative rate. The fluctuations of the table around the bridge are characterized in two complementary ways. Linear statistics satisfy a central limit theorem for bounded test functions, with limiting variance equal to the squared $L^2$-distance, under the bridge, of the test function to the subspace of additively separable functions; as a process they converge to white noise with intensity given by the bridge, conditioned to have vanishing marginals. Spectrally, the limiting spectral distribution of the centered table is governed by a Dyson equation whose variance profile is the density of the bridge. The key ingredient is an asymptotic formula for the partition function of the Gibbs measure in terms of a Fredholm determinant given by the canonical correlations of the bridge. For Mallows-type random permutations, our results prove Pal's conjecture on the limiting partition function and Mukherjee's conjecture on the central limit theorem for linear statistics, both in greater generality.
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