### The random path representation of classical spin systems

Probability Seminar

26th January 2022, 4:00 pm – 5:00 pm

Online,

Many models in statistical mechanics can be written in terms of a collection of (random) geometric objects. This may consist of a "soup" of random walks/loops, random currents, or random transpositions. I will introduce a model of random paths that includes several interesting models when its parameters are chosen appropriately. One such example is the spin O(N) model. Unlike similar representations of this model, the random path model is defined in terms of a product of local terms which allows many nice techniques, such as reflection positivity, to be used. I will show how to use this model to prove exponential decay of correlations for spin O(N) with an external field. The idea of the proof is very simple and applies much more generally than the alternative Lee-Yang method.

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