Models of random subtrees of a graph

Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. A subtree of $G$ with size $n$ is a tree which is a subgraph of $G$, with $n$ vertices. When $n=N$, such a subtree is called a spanning tree. The main purpose of this paper is to explore the question of uniform sampling of a subtree of $G$, or a subtree of $G$ with a fixed number of nodes $n$, for some $n\leq N$. We provide asymptotically exact simulation methods using Markov chains. We highlight the case of the uniform subtree of $\Z^2$ with $n$ nodes, containing the origin $(0,0)$ for which Schramm asked several questions. We produce pictures, statistics, and some conjectures.\par The second aim of the paper is devoted to survey other models of random subtrees of a graph, among them, we will discuss DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide a number of new models, some statistics, and some conjectures.

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