Percolation on Irregular High-dimensional Product Graphs

Abstract

We consider bond percolation on high-dimensional product graphs $G=\square_{i=1}^tG^{(i)}$, where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$, where $d$ is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$, and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev: firstly, we provide a construction showing that the requirement of bounded-degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper.

Figures (5)

• Figure 1: The induction step in Lemma \ref{['t^k']}
• Figure 2: The vertex $u\in Q$ and its exposed neighbourhood in $T_2$, and a vertex $x\in T_2$ and its exposed neighbourhood in $T_1$. Note that in the course of the exploration process, all the vertices in $N_{G_p}(x,T_2)$ move from $T_1$ to $Q$, the vertex $x$ (and, subsequently, all the vertices from $N_{G_p}(u,T_2)$) move to $S_2$, and, finally, the vertex $u$ moves to $S_1$.
• Figure 3: An extension of a partition $A\cup B$ of $W$ to a partition $A'\cup B'$ of $M_{z}$. Each $A'-B'$ two-path (solid) can be extended into an $A-B$ six-path (solid and dashed). Note that some vertices in these paths may lie outside of $M_{z}$.
• Figure 4: A visualisation of the graph $G^2[M_{z}]$ in the case $t=4,z=2$ and $s=2$. Here, each circle is a copy of $H(t-z,s)$ and the black tubes represent the edges of $J(t,z)$. A pair of copies of $H(t-z,s)$ have been displayed, where the vertices of $M_{z}$ inside each copy have been coloured according to whether they lie in $A'$ (blue) or $B'$ (orange). Inside the black tube corresponding to the edge of $J(t,z)$ between this pair of vertices there is some matching, represented by the dashed (purple) edges. Note that this matching may not respect the structure of the $H$ copies in each circle. Each vertex $I$ of $J(t,z)$ is then coloured according to whether it is $A'$-dominated (blue), $B'$-dominated (orange) or evenly balanced (green), corresponding to $\chi(I)=1,2,$ and $3$, respectively.
• Figure 5: A summary of the typical component structure of percolated high-dimensional product graphs

Theorems & Definitions (30)

• Theorem 1: ER60
• Theorem 2: AKS81BKL92
• Theorem 3: L22
• Theorem 6: Informal DEKK22
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10
• Lemma 11
• Theorem 12: CT98