Cotorsion pairs in cluster categories of type $A_{\infty}^{\infty}$

In this paper, we give a complete classification of cotorsion pairs in a cluster category $\mathscr{C}$ of type $A^\infty_\infty$ via certain configurations of arcs, called $\tau$-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify $t$-structures and functorially finite rigid subcategories in $\mathscr{C}$, respectively. We also deduce Liu-Paquette's classification of cluster tilting categories of $\mathscr{C}$ and Ng's classification of torsion pairs in the cluster category of type $A_\infty$.