发表机构
Università degli Studi Roma Tre; Università degli Studi di Milano（罗马第三大学; 米兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机交换模型的谱隙，证明其由至多二次多项式达到，刻画单/双粒子主导条件，解决Alon-Puder猜想，并推广至边界驱动情形。
AI 中文摘要
我们考虑一类广泛的交换动力学，其更新结构为任意加权图或超图，其中包括Kipnis--Marchioro--Presutti（KMP）模型以及Kac在球面上行走的能量。这些是保守的连续自旋系统，其可逆测度为Dirichlet分布。我们证明其谱隙总是由能量变量中次数至多为2的多项式达到。等价地，通过与相关离散粒子系统的交织，主导模态总是由单粒子或双粒子可观测量表示。我们将此解释为Aldous型谱隙现象的一种体现。特别地，由此得到的双粒子谱隙恒等式解决了Alon和Puder最近的一个猜想。我们还精确刻画了何时单粒子足够以及何时真正的双粒子模态占主导。在Dirichlet参数的常见重标度下，段状几何恰好是谱隙总是单粒子型的几何，平均场几何恰好是谱隙总是双粒子型的几何，而所有其他几何则表现出非平凡转变。证明依赖于对所谓隐藏模型（原始过程的对偶表示）的分析。次数二约化的机制非常稳健，并扩展到更广泛的随机交换模型及其相关粒子系统，包括调和过程、即时交换模型、平均型过程以及不可逆变体。最后，我们分析了边界驱动的KMP模型版本，其中体交换动力学与储层相互作用，通常导致不可逆过程。与保守情形相反，我们证明边界驱动模型的谱隙总是单粒子型的。
英文摘要
We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis--Marchioro--Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.
Comments75 pages, 1 figure, notation guide
