TY - JOUR
T1 - Percolation on High-Dimensional Product Graphs
AU - Diskin, Sahar
AU - Erde, Joshua
AU - Kang, Mihyun
AU - Krivelevich, Michael
N1 - Publisher Copyright:
© 2024 The Author(s). Random Structures & Algorithms published by Wiley Periodicals LLC.
PY - 2025/1
Y1 - 2025/1
N2 - We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalizes the results of Ajtai, Komlós, and Szemerédi [1] and of Bollobás, Kohayakawa, and Łuczak [5] who showed that the (Formula presented.) -dimensional hypercube, which is the (Formula presented.) -fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.
AB - We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalizes the results of Ajtai, Komlós, and Szemerédi [1] and of Bollobás, Kohayakawa, and Łuczak [5] who showed that the (Formula presented.) -dimensional hypercube, which is the (Formula presented.) -fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.
KW - percolation
KW - phase transition
KW - product graphs
KW - random graphs
UR - http://www.scopus.com/inward/record.url?scp=85209888576&partnerID=8YFLogxK
U2 - 10.1002/rsa.21268
DO - 10.1002/rsa.21268
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AN - SCOPUS:85209888576
SN - 1042-9832
VL - 66
JO - Random Structures and Algorithms
JF - Random Structures and Algorithms
IS - 1
M1 - e21268
ER -