论文标题
圆环的同态
Homomorphisms from the torus
论文作者
论文摘要
我们提供了来自离散的圆环$ \ mathbb {z} _m^n $的一组加权同构的详细概率和结构分析，其中$ m $甚至是任何固定的图表：我们表明，这些同源物的相应概率分布构成了某些随机危机的构造分布，这些分布构成了某些一定的某些独立型构成的分布。这有几个后果，包括解决方案（以强烈的形式）来猜想英语和加尔文的猜想以及卡恩和帕克的猜想。特殊案例包括针对独立集数量的尖锐渐近学和$ \ m mathbb {z} _m^n $的合适$ q $ - 颜色的数量（尤其是离散的HyperCube）。我们将进一步的应用于高度功能的研究和（广义）等级函数在离散的超立方体上，并反驳了Kahn和Lawrenz的猜想。为了证明，我们结合了来自统计物理，熵和图形容器的方法，并利用圆环的等速度和代数特性。
We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper $q$-colourings of $\mathbb{Z}_m^n$ (so in particular, the discrete hypercube). We give further applications to the study of height functions and (generalised) rank functions on the discrete hypercube and disprove a conjecture of Kahn and Lawrenz. For the proof we combine methods from statistical physics, entropy and graph containers and exploit isoperimetric and algebraic properties of the torus.
