置换群上的随机游动，一些形式的长时间渐近关系

量子Heisenberg铁磁体置换群上无规行走的聚合物展开

A Polymer Expansion for the Random Walk on the Permutation Group Associated to the Quantum Heisenberg Ferromagnet

很长一段时间以来，人们把晶格上的量子海森堡铁磁体与晶格顶点的置换群上的随机游动联系起来。在这里，我们提出了一个聚合物膨胀的热方程的解耦合到随机行走。我们在有限格上工作，不存在收敛问题。我们离开未来的工作，在必要的扩展到一个无限的晶格结果的边界条件。

Scaling limit of a self-avoiding walk interacting with spatial random permutations

We consider nearest neighbour spatial random permutations on $\mathbb{Z}^d$. In this case, the energy of the system is proportional the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $α$, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding wa…

Uniformly positive correlations in the dimer model and phase transition in lattice permutations on $\mathbb{Z}^d$, $d > 2$, via reflection positivity

Our first main result is that correlations between monomers in the dimer model in $\mathbb{Z}^d$ do not decay to zero when $d > 2$. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas.…

大型对称随机游动系统的渐近Feynman-Kac公式

Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks

研究了在对称测度下具有有限时域$ [0,β] $的格点$ \Z^d $上的$ N $随机过程的大偏差原理，其中所有的始点和终点都由一个随机置换一致给定.也就是说，给定$ N $元素的置换$ σ$和$ N $初始点的向量$ (x_1,...,x_N) $，我们让随机过程终止于点$ (x_{σ(1)},...,x_{σ(N)}) $，然后对所有可能的置换和初始点求和，用初始分布加权。有一个两级随机机制，我们证明了两级大偏差原则的平均经验路径措施，平均路径和平均占领当地时间在此对称措施。对称测度不能写成任何单个随机过程分布的乘积。我们展示了这些结果在量子统计力学中的几个重要应用，使用Feynman-Kac公式表示某些迹类算子的迹。特别地，我们证明了具有玻色-爱因斯坦统计和平均场相互作用的量子自旋系统的非对易Varadhan引理。对于$ N $简单随机游动的局部时的平均值，我们的大偏差原理的一个特例是对于有限时间$ β$，但对于有限时间$ N\to\infty $，Donsker-Varadhan速率函数作为速率函数.对这一令人惊讶的结果给出了量子统计力学的解释。

多项式李群积分的曲面映射展开及置换的因式分解

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

利用高斯随机矩阵积分的图解方法，我们找到了多项式李群积分在曲面上的无限和表示。所涉及的映射满足一个特定的条件：它们有一些标记的顶点，并且没有避开这些顶点的闭合行走。我们还制定我们的结果的排列，到达新的类型的因式分解问题。

Gaussian random permutation and the boson point process

We construct an infinite volume spatial random permutation $(\mathsf X,σ)$, where $\mathsf X\subset\mathbb R^d$ is locally finite and $σ:\mathsf X\to \mathsf X$ is a permutation, associated to the formal Hamiltonian $$ H(\mathsf X,σ) = \sum_{x\in \mathsf X} \|x-σ(x)\|^2. $$ The measures are parametrized by the point density $ρ$ and the temperature $α$. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Let $ρ_c=ρ_c(α)$ be the critical density for Bose-Einstein condensation in Feynman's representation. Each finite cycle …