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arXiv · 2512.13418

Large Deviation Properties of Minimum Spanning Trees for Random Graphs

Mahdi Sarikhani·Alexander K. Hartmann
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Abstract

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$ conditioned to be connected. By using large-deviation Markov chain sampling, we are able to obtain the distribution $P(W)$ of the spanning-tree weight $W$ down to probability densities as small as $10^{-300}$. For the complete graph, we confirm analytical predictions with respect to the expectation value. For both ensembles, the large deviation principle is fulfilled. For the connected ER graphs, we observe a remarkable change of the distributions at the value of $c=1$, which is the percolation threshold for the original ER ensemble.

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Mahdi Sarikhani, Alexander K. Hartmann. 2025-12-15. Large Deviation Properties of Minimum Spanning Trees for Random Graphs. https://doi.org/10.1103/xc5g-t9dj

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