AI 中文总结
本研究利用自旋1/2 XX链的基态变换表示，推导对称Dyson排斥过程的精确演化公式，明确其密度矩层级由有限维多项式代数支配，所得结果与推测的流体动力学结论一致。
AI 中文摘要
对称Dyson排斥过程（SDEP）是具有长程对数库仑型相互作用的格点排斥过程，存在多种等价形式：可视为Doob变换意义下受约束不碰撞的对称随机游走、受约束简单对称排斥过程（SSEP）的最大活性极限，或自旋1/2 XX链的基态变换。本研究利用后一种表示，从确定性初始构型推导得到精确演化公式，所得行列式核通过拉格朗日多项式的有限插值表达式表示，可给出有限与无限格点上的显式密度演化。对于密堆积块的熔化过程，研究发现密度矩的完整层级由有限维多项式代数支配，且在主导时间系数中识别出卡特兰数；还推导了欧拉尺度密度分布与分离冻结区和液态区的北极曲线，证明其与前期工作中推测的流体动力学结果一致。
英文摘要
The symmetric Dyson exclusion process (SDEP) is an exclusion process on the lattice with a long-range logarithmic Coulomb-type interaction. It appears in several equivalent forms: as symmetric random walkers conditioned, in the Doob-transform sense, not to collide; as the maximal-activity limit of a conditioned SSEP; and as a ground-state transform of the spin- 1/2 XX chain. In this work, we exploit this latter representation to obtain exact evolution formulas from deterministic initial configurations. The resulting determinantal kernel is expressed through a finite interpolation expression in terms of Lagrange polynomials, leading to explicit density evolution in finite and infinite lattices. For the melting of a densely packed block, we show that the full hierarchy of density moments is governed by a finite-dimensional polynomial algebra, and we identify Catalan numbers in the leading time coefficients. We also derive the Euler-scale density profile and the arctic curve separating frozen and liquid regions, and show that they coincide with conjectured hydrodynamic results obtained in a previous work.
