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arXiv · 2304.09815

On a generalised Lambert $W$ branch transition function arising from $p,q$-binomial coefficients

Per Åhag·Rafał Czyż·Per-Håkan Lundow
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Abstract

With only a complete solution in dimension one and partially solved in dimension two, the Lenz-Ising model of magnetism is one of the most studied models in theoretical physics. An approach to solving this model in the high-dimensional case ($d>4$) is by modelling the magnetisation distribution with $p,q$-binomial coefficients. The connection between the parameters $p,q$ and the distribution peaks is obtained with a transition function $\omega$ which generalises the mapping of Lambert $W$ function branches $W_0$ and $W_{-1}$ to each other. We give explicit formulas for the branches for special cases. Furthermore, we find derivatives, integrals, parametrizations, series expansions, and asymptotic behaviors.

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Per Åhag, Rafał Czyż, Per-Håkan Lundow. 2023-04-19. On a generalised Lambert $W$ branch transition function arising from $p,q$-binomial coefficients. https://arxiv.org/abs/2304.09815

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