arXiv subjects

Kelvyn Welsch

Publications and source records attributed to Kelvyn Welsch.

3 recordsLinked to original sources

arXiv2025

Phase Transition in Long-Range $q-$state Models via Contours. Clock and Potts Models with Fields

Using the group structure of the state space of $q-$state models, a new definition of contour for long-range spin-systems in $\Z^d$ ($d\geq 2$), and a multidimensional version of Fröhlich-Spencer contours, we prove phase transition for a class of ferromagnetic long-range systems which includes the Clock and Potts models. Our arguments work for the entire region of exponents of regular power-law interactions, namely $α> d$, and for any $q \geq 2$. As an application, we prove phase transition for Potts models with decaying fields when the field decays fast enough and in the presence of a random external field.

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arXiv2025

Cluster Expansion and Decay of Correlations for Multidimensional Long-Range Ising Models

We develop the cluster expansion for the multidimensional multiscaled contours defined by three of us. These contours are suitable for long-range Ising models with interaction $J_{xy}=J(|x-y|)= J/|x-y|^α$, $J>0$, and $α>d$. As an application of the convergence of the cluster expansion at low temperatures, we study the decay of the truncated two-point correlation functions, showing that the decay is algebraic with coefficient $α$.

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arXiv2024

Phase Transitions on 1d Long-Range Ising Models with Decaying Fields: A Direct Proof via Contours

Following seminal work by J. Fröhlich and T. Spencer on the critical exponent $α=2$, we give a proof via contours of phase transition in the one-dimensional long-range ferromagnetic Ising model in the entire region of decay, where phase transition is known to occur, i.e., polynomial decay $α\in (1,2]$. No assumptions that the nearest-neighbor interaction $J(1)$ is large are made. The robustness of the method also yields a proof of phase transition in the presence of a nonsummable external field that decays sufficiently fast.

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