Related papers: Uniqueness and multiplicity of infinite clusters
We consider isomorphism properties of infinite random geometric graphs defined over a variety of metrics. In previous work, it was shown that for $\mathbb{R}^n$ with the $L_{\infty}$-metric, the infinite random geometric graph is, with…
An urn scheme is a probabilistic model in which balls are placed into urns sequentially and independently of each other. All balls share the same probability distribution for hitting the urns. In the simplest case, there is a finite number…
In this note we make progress toward a conjecture of Durham--Fanoni--Vlamis, showing that every infinite-type surface with finite-invariance index 1 and no nondisplaceable compact subsurfaces fails to have a good curve graph, that is, a…
Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian…
We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this…
Indistinguishability of particles is normally considered to be an inherently quantum property which cannot be possessed by a classical theory. However, Saunders has argued that this is incorrect, and that classically indistinguishable…
We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration ("pattern") of sites and bonds can occur in large clusters, then it occurs at…
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…
In a recent paper [1], it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign),…
We study a uniform, quantitative form of the amenability-hyperfiniteness paradigm for bounded-degree Borel graphs generating countable Borel equivalence relations. We introduce \emph{uniform Borel amenability} and prove that it is…
We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size…
Clustering is a widely used unsupervised learning method for finding structure in the data. However, the resulting clusters are typically presented without any guarantees on their robustness; slightly changing the used data sample or…
An infinite urn scheme is defined by a probability mass function $(p_j)_{j\geq1}$ over positive integers. A random allocation consists of a sample of $N$ independent drawings according to this probability distribution where $N$ may be…
Birman-Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of Birman-Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid…
In cluster analysis interest lies in probabilistically capturing partitions of individuals, items or observations into groups, such that those belonging to the same group share similar attributes or relational profiles. Bayesian posterior…
Recently, Buan and Marsh showed that if two complete $\tau$-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is $\tau$-tilting finite. They conjectured that the result holds…
We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…
This paper consists of three steps. In the first, we prove that the Bekenstein-Hawking entropy is the unique expression of black hole entropy. Our proof is constructed in the framework of thermodynamics without any statistical discussion.…