The matroid of a graphing (Q6615766)

Graphings are limit objects for bounded-degree graphs. The ``cycle matroid'' of a graphing is defined as a submodular set function, with values in the closed interval \([0,1]\). It generalizes the cycle matroid of finite graphs in the following way: For a finite graph \(G=(V,E)\), we define the normalized rank function \(\rho(X)\) by \(\rho(X)=r(X)/|V|\), where \(r(X)\) is the usual rank function. Let \(\mathcal{P}\) be the partition of the node set \(V\) into the connected components of subgraph \((V,X)\). For \(v\in V\), let \(\mathcal{P}_v\) denote the partition class containing \(v\). Let \(\mathfrak{u}\) be a uniform random node of \(V\), then \N\[\N\mathcal{E}\bigg(\frac{1}{|\mathcal{P}_{\mathfrak u}|}\bigg)=\frac{|\mathcal{P}|}{ |V|}=1-\rho(X).\N\]\NIn order to extend to the graphing case, let \(\mathcal{G}=(J, \mathcal{B}, E, \lambda)\) be a graphing. For every Borel set \(X\subseteq E\), the quadruple \(\mathcal{G}^X=(J, \mathcal{B}, X, \lambda)\) is a graphing. We denote the partition of \(J\) into the connected components of \(\mathcal{G}^X\) by \(\mathcal{P}^X\), we then define \N\[\N\rho(X)=\rho_{\mathcal{G}}(X)=1-\mathcal{E}_{\mathfrak{u}}\bigg(\frac{1}{|V(G_{\mathfrak{u}^X}|)}\bigg)=1-\psi(\mathcal{P}^X), \text{ where }\psi(\mathcal{P})=\mathcal{E}\bigg(\frac{1}{|\mathcal{P}_{\mathfrak{u}}|}\bigg),\N\] \Nfor \(\mathfrak{u}\) chosen from the distribution \(\pi\). \N\NFurthermore, for every Borel set \(X\subseteq E\), we define the edge measure by \N\[\N\eta(X)=\int_J \deg_X(u)\ d\lambda(u).\N\]\NThe three major results of the paper are summarized as follows: \N\NTheorem 4.2. The set function \(\rho\) defined on the Borel subset of \(E\) is increasing and submodular, and it satisfies the inequalities: \N\[\N\frac{1}{1+D}\eta (X)\le \rho (X)\le \eta(X).\N\]\NTheorem 4.4. Let \(\mathcal{G}\) be a hyperfinite graphing and \(\mathcal{H}=(J,F)\), a Borel measurable spanning subforest of \(G\), then\N\[\N\alpha(X)=\frac{1}{2} \eta_{\mathcal{G}}(X\cap F)\N\]\Ndefines an extremal maximal minorizing measure on the Borel subsets of \(E\). \N\NTheorem 4.6. If \(G_1, G_2, \dots\) is a sequence of finite graphs with all degrees bounded by \(D\ge 0\) locally converging to a graphing \(\mathcal{G}=(J, \mathcal{B}, E, \lambda)\), then \(\bar \rho(G_n)\rightarrow \bar \rho(\mathcal{G})\) as \(n\rightarrow \infty\).