**Part 3: Jesper Lykke Jacobsen ‘High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials’**

Since 2009, the *Journal of Physics A* has awarded a Best Paper Prize, which serves to celebrate and applaud well written papers that make a significant contribution to their field. In 2016, the Editorial Board awarded 3 prizes using the criteria of novelty, achievement, potential impact and presentation. In the final of a series of three interviews, meet Jesper Lykke Jacobsen, the author of ‘__High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials__’ Jesper Lykke Jacobsen 2014 *J. Phys. A: Math. Theor.* **47** 135001*.*

### Meet the author

Jesper Jacobsen prepared his Ph.D. in physics at Oxford University and Aarhus University. He is presently a Professor at the University Pierre and Marie Curie and the Ecole Normale Supérieure in Paris.

### What prompted you to pursue this field of research?

One key question in the field of exact solutions and integrability is what property makes a model exactly solvable. Baxter and collaborators have solved the Q-state Potts model on the square and triangular lattices, along certain critical manifolds. On the other hand, the same model on the kagome lattice does not appear to be solvable, although an accurate approximation to its critical manifold was provided by Wu.

With my collaborator C.R. Scullard we have defined a topologically weighted graph polynomial whose zeros give the exact critical manifold in solvable cases of Potts, Ising and percolation models. In the non-solved (and presumably non-solvable) cases, the zeros provide very good approximations to the critical manifold. The accuracy improves remarkably fast upon increasing the size of its ‘basis’, *i.e*., the finite piece of the lattice on which it is computed. I wanted to see if the factorisation properties of the graph polynomial could serve to detect new cases of solvability

### What is the winning paper all about?

I found an efficient way to compute this graph polynomial, by combining knowledge about transfer matrices and the representation theory of the Temperley-Lieb algebra. I then pushed the computations to as large bases as possible, on a score of standard lattices including the Archimedean ones. This has shed light on the critical manifolds of all these models, and in particular the antiferromagnetic region turns out to harbour a very rich phase diagram. The factorisation property has also identified a handful of new cases which are very likely exactly solvable. Finally, the ferromagnetic critical points and percolation thresholds have been determined to a very high numerical precision, largely exceeding that of more standard numerical simulations.

### What do you plan to do next?

A natural next step was to transform the graph polynomial into an eigenvalue problem, while maintaining the factorisation property. I already succeeded in doing this (J. Phys. A: Math. Theor. 48, 454003 (2015), and I also found a generalisation to the O(n) model. I now hope to define the O(n) model graph polynomial for finite bases. It is also natural to try to understand better the meaning of the graph polynomial from the perspective of integrability, representation theory, and discrete holomorphicity.

To see the rest of the winners and the nomination process for the *Journal of Physics A* Best Paper Prize 2017, visit this page.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.

[1] © Jesper Lykke Jacobsen.

Categories: Journal of Physics A: Mathematical and Theoretical