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Comparison of the random loop model to percolation and infinite loops in the random link model

Peter Mühlbacher showed that the inverse temperature of the random loop model can be compared to a percolation parameter which leads to a bound where infinitely large loops do not occur. By a different approach, we improve the bound for the inverse temperature. We also obtain a numerical value such that the inverse temperature must be chosen larger than it in order to obtain infinitely long loops. Furthermore, we demonstrate how this comparison argument can be applied to compare the random link model to percolation. This is a joint work with Benjamin Lees and Mino Nicola Kraft.