Random subgraphs of finite graphs. I: The scaling window under the triangle condition.

*(English)*Zbl 1076.05071Let \(G\) be a finite connected transitive graph with \(V\) vertices. Transitive: For every pair of vertices \(x\), \(y\) there is a graph isomorphism \(\varphi\) with \(\varphi(x)= y\). Then every vertex has the same degree \(\Omega\). The edges of \(G\) are deleted independently with probability \(1-p\), so that a stochastic subgraph \(H\) of \(G\) results. Let \(\tau_p(x,y)= P(x\leftrightarrow y)\) where \(x\leftrightarrow y\) means that there is a path from \(x\) to \(y\) in \(H\). Let \(C(x)\) be the cluster of vertex \(x\), i.e. \(C(x)= \{y\in G: x\leftrightarrow y\}\). The distribution of \(|C(x)|=\) cardinality of \(C(x)\) is independent of \(x\). Put \(\chi(p)= E|C(0)|= \sum_x\tau_p(0,x)\) (susceptibility). Dependence on \(p\) and \(V\) is sometimes omitted.

The paper studies the phenomenon that in a (small) neighbourhood (the window) of a suitable probability \(p_c\) (the critical threshold) the behaviour of \(\chi(p)\), \(E|C_{\max}|= E\max\{|C(x)|:x\) vertex} and related quantities for large \(V\) is different from the behaviour above and below the window (the latter also different). This is classical for the random graph, i.e. for \(H\) when \(G\) is complete. The threshold \(p_c\) is taken as the unique solution \(p\) of \(\chi(p)=\lambda V^{1/3}\) for some small constant \(\lambda> 0\), a choice based on known behaviour of the random graph. Some variation of \(p_c\) in a window is allowed. A typical window is \([p_c- \varepsilon\Omega^{-1}, p_c+ \varepsilon\Omega^{-1}]\), \(0< \varepsilon\leq\Lambda V^{-1/3}\), \(\Lambda\) constant.

Throughout the paper the authors assume the finite-graph triangle condition \(\Delta(p_c, x,y)\leq \delta(x,y)+ a_0\) for a sufficiently small constant \(a_0\) , where \(\Delta(p,x,y)= \sum\tau_p(x,u)\tau_p(u, v)\tau_p(v,y)\) with sum over all vertices \(u\), \(v\) of \(G\). General results on the above behaviour are proved: bounds on \(p_c\) and bounds for \(\chi(p)\) and \(E|C_{\max}|\) in, below and above the window. Also for percolation probability \(P(|C(0)|\geq N)\), \(N= \varepsilon^{\alpha- 2} V^{\alpha/3}\), \(0<\alpha< 1\), and the magnetization \(1-\sum_{k\leq V}(1- \gamma)^k P(|C(0)|= k)\), \(0< \gamma< 1\).

It is shown that the triangle condition holds for the random graph. This implies theorems approximating known results for the random graph. Other special cases are tori where the authors invoke their paper [Random subgraphs of finite graphs. II: The lace expansion and the triangle condition. Ann. Probab., in press].

Proofs make use of the differential inequality \([1-\max\Delta(p,x,y)]\Omega\leq d\psi/dp\leq \Omega\), where \(\max\) is over all edges \([x, y]\) in \(G\) and \(\psi= 1/\chi\). Bounds on the magnetization, that imply estimates for \(P(C(0)\geq k)\), are derived by other differential inequalities.

The paper studies the phenomenon that in a (small) neighbourhood (the window) of a suitable probability \(p_c\) (the critical threshold) the behaviour of \(\chi(p)\), \(E|C_{\max}|= E\max\{|C(x)|:x\) vertex} and related quantities for large \(V\) is different from the behaviour above and below the window (the latter also different). This is classical for the random graph, i.e. for \(H\) when \(G\) is complete. The threshold \(p_c\) is taken as the unique solution \(p\) of \(\chi(p)=\lambda V^{1/3}\) for some small constant \(\lambda> 0\), a choice based on known behaviour of the random graph. Some variation of \(p_c\) in a window is allowed. A typical window is \([p_c- \varepsilon\Omega^{-1}, p_c+ \varepsilon\Omega^{-1}]\), \(0< \varepsilon\leq\Lambda V^{-1/3}\), \(\Lambda\) constant.

Throughout the paper the authors assume the finite-graph triangle condition \(\Delta(p_c, x,y)\leq \delta(x,y)+ a_0\) for a sufficiently small constant \(a_0\) , where \(\Delta(p,x,y)= \sum\tau_p(x,u)\tau_p(u, v)\tau_p(v,y)\) with sum over all vertices \(u\), \(v\) of \(G\). General results on the above behaviour are proved: bounds on \(p_c\) and bounds for \(\chi(p)\) and \(E|C_{\max}|\) in, below and above the window. Also for percolation probability \(P(|C(0)|\geq N)\), \(N= \varepsilon^{\alpha- 2} V^{\alpha/3}\), \(0<\alpha< 1\), and the magnetization \(1-\sum_{k\leq V}(1- \gamma)^k P(|C(0)|= k)\), \(0< \gamma< 1\).

It is shown that the triangle condition holds for the random graph. This implies theorems approximating known results for the random graph. Other special cases are tori where the authors invoke their paper [Random subgraphs of finite graphs. II: The lace expansion and the triangle condition. Ann. Probab., in press].

Proofs make use of the differential inequality \([1-\max\Delta(p,x,y)]\Omega\leq d\psi/dp\leq \Omega\), where \(\max\) is over all edges \([x, y]\) in \(G\) and \(\psi= 1/\chi\). Bounds on the magnetization, that imply estimates for \(P(C(0)\geq k)\), are derived by other differential inequalities.

Reviewer: A. J. Stam (Winsum)

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