We introduce a new stick-breaking construction for inhomogeneous continuum random trees (ICRT). This new construction allows us to prove the necessary and sufficient condition for compactness conjectured by Aldous, Miermont and Pitman [1] by comparison with Lévy trees. We also compute the fractal dimensions (Minkowski, Packing, Hausdorff).

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Motivated by the scaling limits of the connected components of the configuration model, we study uniform connected multigraphs with fixed degree sequence D and with surplus k. We call those random graphs (D, k)-graphs. We prove that, for every k ∈ N, under natural conditions of convergence of the degree sequence, (D, k)-graphs converge toward either (P, k)-graphs or (Θ, k)-ICRG (inhomogeneous continuum random graphs). We prove similar results for (P, k)-graphs and (Θ, k)-ICRG, which have applications to multiplicative graphs. Our approach relies on two algorithms, the cycle-breaking algorithm, and the stick-breaking construction of D-tree that we introduced in a recent paper. From those algorithms we deduce a biased construction of (D, k)-graph, and we prove our results by studying this bias.

We introduce a new theory of plane R-tree, to define plane ICRT (inhomogeneous continuum random tree), and its looptree, fennec (a Gaussian free field on the looptree), and snake. We prove that a.s. the looptree is compact, and that a.s. the fennec and snake are continuous. We compute the looptree's fractal dimensions, and the fennec and snake's Hölder exponent. Alongside, we define a Gaussian free field on the ICRT, and prove a condition for its continuity. In a companion paper [8], we prove that the looptrees, fennecs, and snakes of trees with fixed degree sequence converge toward the looptrees, fennecs and snakes of ICRT.