The Loop O(n) Model

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What is this?

The picture is a sample from the loop O(n) model on a 340x300 rectanglur portion of the hexagonal lattice. A configuration in the model consists of a collection of disjoint loops. Not all configurations in the model are equally likely, with the probability of any given configuration depending on the number N of loops in it and the total length M of the loops in it. The probability of a configuration is proportional to n^Nx^M, where n>0 and x>0 are two parameters of the model. The picture shows a random configuration chosen from this model with parameters n=0.5 and x=0.6 (the longest loops are colored red, blue, green, purple, orange, ...).

Mathematics

What are the possible behaviors of the model? That is, in a typical configuration, how many loops are there and what are their lengths? Some possibilities are: (i). There are many long loops. (ii) There are many loops, but they are all rather short. (iii) There are few loops and they are all rather short. What determines the behavior? The two parameters of the model, n and x, together determine the behavior, though it is not an easy matter to figure out which values of n and x lead to which type of behavior. A first observation is that when n<1, configurations with many loops are less likely, whereas when n>1, they are more likely. Similarly, when x<1, configurations with a large total length of loops are less likely, whereas when x>1, they are more likely. However, since there are many possible configurations, there are competing forces at play, and this does not immediately lead to an answer. For example, when n>2, it is expected that behavior (i) never occurs. This is partially resolved in [2], where it is shown that when n and x are both large, behavior (ii) takes place, whereas when x is small, behavior (iii) takes place. When n≤2 and x is at least a certain "critical" threshold, it is expected that behavior (i) occurs. This is shown in [1] when 1≤ n≤2 and x precisely equals this critical value. See [3] for more results and discussion on this model.

References

[1] H. Duminil-Copin, A. Glazman, R. Peled and Y. Spinka. Macroscopic loops in the loop O(n) model at Nienhuis' critical point. Journal of the European Mathematical Society, 2020.

[2] H. Duminil-Copin, R. Peled, W. Samotij and Y. Spinka. Exponential decay of loop lengths in the loop O(n) model with large n. Communications in Mathematical Physics, 2017.

[3] R. Peled and Y. Spinka. Lectures on the Spin and Loop O(n) Models. Sojourns in Probability and Statistical Physics, 2019.