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Abstract We determine the order of thek-core in a large class of dense graph sequences. Let$$G_n$$be a sequence of undirected,n-vertex graphs with edge weights$$\{a^n_{i,j}\}_{i,j \in [n]}$$that converges to a graphon$$W\colon[0,1]^2 \to [0,+\infty)$$in the cut metric. Keeping an edge (i,j) of$$G_n$$with probability$${a^n_{i,j}}/{n}$$independently, we obtain a sequence of random graphs$$G_n({1}/{n})$$. Using a branching process and the theory of dense graph limits, under mild assumptions we obtain the order of thek-core of random graphs$$G_n({1}/{n})$$. Our result can also be used to obtain the threshold of appearance of ak-core of ordern.
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This content will become publicly available on June 1, 2026
Irrational rotations and $2$-filling rays
Abstract We study a skew product transformation associated to an irrational rotation of the circle$$[0,1]/\sim $$. This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of$$\{0,1/2\}$$in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is aninfiniteclique of$$2$$-filling rays based at that end. These$$2$$-filling rays are relevant to Bavard and Walker’sloop graphs.
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- Award ID(s):
- 2202986
- PAR ID:
- 10638489
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 45
- Issue:
- 6
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1673 to 1697
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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