Matchings in Benjamini-Schramm convergent graph sequences
Miklos Abért, Péter Csikvári, Péter Frenkel, Gábor Kun
备注：
21 pages. Changes compared to the previous (published) version: In Lemma 4.3, formula (4.3) and its proof have been corrected. As a consequence, the formula for the variance in Theorem 4.5 has been modified
We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree.
A graph parameter is said to be estimable if it converges along every Benjamini-Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d-regular bipartite graphs for any fixed d at least 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge.
However, for any sequence of d-regular bipartite graphs converging to the d-regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d-regular bipartite graphs. We show that the limit equals to the exponent in Schrijver's lower bound on the number of perfect matchings.
Our analytic approach also yields a short proof for the Nguyen-Onak (also Elek--Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.
Benjamini-Schramm limit of the heat semigroup on quantum graphs
Department of Differential Equations, Faculty of Natural Sciences, Budapest University of Technology and Economics（布达佩斯技术与经济大学自然科学学院微分方程系）；Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
Benjamini，Kalai and Schramm（2001）证明了$n$独立无偏比特的加权多数函数在噪声下是一致稳定的：当每个比特以概率$ε$翻转时，加权多数改变的概率$p_ε$至多为$Cε^{1/4}$。他们问什么是可以取代1/4的最佳指数。我们证明答案是1/2。$p_ε$的上界与$ε\to 0$和$nε\to \infty$的已知下界相差$\sqrt{π/2}+o(1)$。
We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infi…
