Svante Janson
We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.
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We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability.
S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probabili…
We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields c…
In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right $n/2\times n/2$ submatrix. The result has an application to estimating the spectral gap of random undirected $d$-regular graphs in terms of the second singular value of {\it directed} random graphs with predefined degree sequences.
Exchangeable pairs, switchings, and random regular graphs
本文利用 Stein 可交换对方法证明随机正则图环数的 Poisson 近似并给出总变差显式界，进而得到线性特征值泛函的极限分布，同时揭示其与 switchings 方法的联系。
De Finetti's classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1,2,3], Hoover [41,42], Kallenberg [44] and Kingman [47]. On the other hand, such exchangeable laws were first related to questions from combinatorics in an independent analysis by Fremlin and Talagrand [29], and again more recently in Tao [62], where they appear as a natural proxy for the `leading order statistics' of colourings of large graphs or hypergraphs. Moreover, this…
We consider a process of noncolliding $q$-exchangeable random walks on $\mathbb{Z}$ making steps $0$ (straight) and $-1$ (down). A single random walk is called $q$-exchangeable if under an elementary transposition of the neighboring steps (down, straight) $\to$ (straight, down) the probability of the trajectory is multiplied by a parameter $q\in(0,1)$. Our process of $m$ noncolliding $q$-exchangeable random walks is obtained from the independent $q$-exchangeable walks via the Doob's $h$-transform for a certain nonnegative eigenfunction $h$ with the eigenvalue less than $1$. The system of $m$ w…
The Critical Beta-splitting Random Tree III: The exchangeable partition representation and the fringe tree
本文研究临界beta分裂随机树的渐近性质，提出规范嵌入连续时间模型与显式刻画随机叶子对应的极限边缘分布两种概率方法，为该模型的结构分析提供新工具。
In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in $\mathbb{R}_+$. Here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over $σ$-finite measure spaces, we can model a large family of exchangeable graphs, including the Caron-Fox graphs and the traditional exchangeable dense graphs as special cases. Explicitly, modelling the underlying space of features by a $σ$-finite measure space $(S,\mathcal{S},μ)$ and the connection pr…
For the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edges essentially matching results for Erdős-Rényi digraphs obtained from Chernoff-type bounds. The proofs make use of the method of exchangeable pairs, developed for concentration of measure by Chatterjee. Exchangeable pairs are constructed using two involutions on the set of regular digraphs: a well-known "simple switching" operation, as well a…
We propose a simple algorithm generating labelled posets of given size according to the almost uniform distribution. By "almost uniform" we understand that the distribution of generated posets converges in total variation to the uniform distribution. Our method is based on a Markov chain generating directed acyclic graphs.
We introduce a class of random graphs that we argue meets many of the desiderata one would demand of a model to serve as the foundation for a statistical analysis of real-world networks. The class of random graphs is defined by a probabilistic symmetry: invariance of the distribution of each graph to an arbitrary relabelings of its vertices. In particular, following Caron and Fox, we interpret a symmetric simple point process on $\mathbb{R}_+^2$ as the edge set of a random graph, and formalize the probabilistic symmetry as joint exchangeability of the point process. We give a representation th…
Extremal persistence probabilities of exchangeable sign-invariant random variables
该研究确定了可交换且符号不变随机变量部分和序列弱与强持续概率的最优上下界，证明两者均以 $n^{-1/2}$ 阶衰减，并补充了确定性情形及允许取零值的相关结果。
Let $\mathcal P(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal P (n,p)$ denote the random poset obtained from $\mathcal P(n)$ by retaining each element from $\mathcal P (n)$ independently at random with probability $p$ and discarding it otherwise. Given any fixed poset $F$ we determine the threshold for the property that $\mathcal P(n,p)$ contains $F$ as an induced subposet. We also asymptotically determine the number of copies of a fixed poset $F$ in $\mathcal P(n)$. Finally, we obtain a number of results on the Ramsey properties of the random poset $\mathcal P(n,p…
Topological expansion for posets and the homological $k$-connectivity of random $q$-complexes
本文针对超越单纯复形的高维展开问题，开发了新的拓扑技术，证明完全q-复形的相关性质，建立随机q-复形连通性的尖锐阈值，还解决了一项同调猜想。
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
For two collections of nonnegative and suitably normalised weights $\W=(\W_j)$ and $\V=(\V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,...,n\}$ is defined by assigning to a generic partition $\{A_j, j\leq k\}$ the probability $\V_{n,k} \W_{|A_1|}... \W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $Π_n$ of $[n]$ are consistent as $n$ varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights $\W$ mu…
In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-…
The automorphism group of the random poset does not admit a generic pair
该研究针对Truss提出的问题，证明随机偏序集的每一对自同构的共轭类都是贫集，同时说明当前定理2的证明存在缺漏，研究仍在推进中。
Entropy of Exchangeable Random Graphs
本文提出一套 graphon 熵估计器以量化可交换随机图的复杂度，建立其收敛速率与中心极限定理，并通过仿真和实际应用验证其有效性。
We give a description of a new Macaulay2 package called SimplicialPosets. This package provides functions for working with simplicial posets and calculating their generalized Stanley-Reisner ideals. For practical purposes, we also introduce of a new random model for a class of simplicial posets which generalizes existing models for random simplicial complexes such as the Kahle model.
Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give…
Previously, Erdős, Kierstead and Trotter investigated the dimension of random height~$2$ partially ordered sets. Their research was motivated primarily by two goals: (1)~analyzing the relative tightness of the Füredi-Kahn upper bounds on dimension in terms of maximum degree; and (2)~developing machinery for estimating the expected dimension of a random labeled poset on $n$ points. For these reasons, most of their effort was focused on the case $0<p\le 1/2$. While bounds were given for the range $1/2\le p <1$, the relative accuracy of the results in the original paper deteriorated as $p$ approa…
We examine properties of generic automorphisms of the random poset, with the goal of explicitly characterizing them. We associate to each automorphism an auxiliary first-order structure, consisting of the random poset equipped with an infinite sequence of binary relations which encode the action of the automorphism. We then explicitly characterize generic automorphisms in terms of properties of this structure. Two notable such properties are ultrahomogeneity, and universality for a certain class of finite structures in this language. As this auxiliary structure seems to be new, we also address…
Intersection statistics for antichains in minuscule posets
研究极小偏序集中抗链交集的统计特性，通过闭式表达揭示其与权重图的联系。
Dimension of unicycle posets
受随机偏序集维数研究的启发，Bollobás和Brightwell于1997年猜想：若有限偏序集的覆盖图至多含一个环，则其序维数至多为3。本文通过构造性证明给出显式的线性扩展三元组，证实该猜想。
A grid poset -- or grid for short -- is a product of chains. We ask, what does a random linear extension of a grid look like? In particular, we show that the average "jump number," i.e., the number of times that two consecutive elements in a linear extension are incomparable in the poset, is close to its maximum possible value. The techniques employed rely on entropy arguments. We finish with several interesting questions about this wide-open area.
We study the relationship between the underlying structure of posets and the spectral and combinatorial properties of their higher-order random walks. While fast mixing of random walks on hypergraphs has led to myriad breakthroughs throughout theoretical computer science in the last five years, many other important applications (e.g. locally testable codes, 2-2 games) rely on the more general non-simplicial structures. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the o…
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