Guttmann 2015 - 70 and counting

The 8th order linear ODEs for the perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons have been known for 10 years. We analyze them in detail, and breaking them up into sums of products of operators up to 3rd order, we find exact closed-form solutions to all operators in terms of 2F1 hypergeometric functions. This is the first time a non-algebraic yet D-finite solution has been demonstrated for self-avoiding polygon generating functions. We further find that the three generating functions differ by the sum of simple algebraic functions and all three are jointly solutions of an order 12 linear ODE.

The first time I really noticed the Heun equation was probably in talks by Tony Guttmann, where it appeared in the the enumeration of staircase polygons and the calculation of lattice Green functions. The Heun equation is a second order linear differential equation with four regular singular points. By confluence it generates four different differential equations: the confluent, double confluent, biconfluent and triconfluent Heun equations. The various cases of this equation have appeared in many applications in the physical sciences. Most recently the confluent Heun equation has been seen to be central to the solution of the quantum Rabi model, which is the basic model for quantum light interacting with matter. The quantum Rabi model is relevant to fundamental experiments in solid state and optical physics, ranging from circuit to cavity quantum electrodynamics. In this talk I will discuss the solution of this model in terms of confluent Heun functions, along with related developments.

I'll discuss some solvable models of polymer adsorption in two dimensions, with a particular focus on a "non-directed" model based on prudent walks. The adsorption transition for these objects turns out to be -- somewhat unexpectedly -- first-order, but recent work by Tony Guttmann and collaborators may help to clarify the matter.

I shall describe five encounters over nearly 30 years with Sloane's (Online) Encylopedia of Integer Sequences.

We construct bosonic solutions to the Zamolodchiov-Faddeev algebra from solutions of the Yang-Baxter equation. These solutions give rise to new expressions for Macdonald polynomials in the form of matrix product formulas. More generally our method provides matrix product formulas for stationary state probabilities of inhomogeneous multi-species asymmetric exclusion processes.

Several decades of parallel developments in the calculation and analysis of series expansions for lattice statistics have led to many new insights into critical phenomena. These studies have centered on the use of the finite lattice method for series expansions in lattice statistics and the use of differential approximants in analysing such series. One of these strands of research ultimately led to the result that a number of unsolved lattice statistics problems cannot be expressed as D-finite functions. Somewhat ironically, given power and success of differential approximants in analysing series, neither the assumed functional form, nor any finite generalisation thereof can fit such cases exactly.

We consider the worm process for the zero-field ferromagnetic Ising model, introduced by Prokofiev and Svistunov. We prove the process is rapidly mixing on all finite graphs and at all temperatures. As a corollary, we construct fully-polynomial randomized approximation schemes for the Ising susceptibility and two-point correlation function.Coauthors: A. Collevecchio, T. Hyndman and D. Tokarev

We revisit the problem of proper vertex colourings of the triangular lattice, using Q distinct colours. Our approach is based on a mapping onto an integrable spin-one model, which can be solved exactly using Bethe Ansatz techniques. In particular we focus on the low-energy excitations above the eigenlevel g_2, which was shown by Baxter (in 1986) to dominate the transfer matrix spectrum in the Fortuin-Kasteleyn (chromatic polynomial) representation for Q_0 <= Q <= 4, where Q_0 = 3.819671... We argue that g_2 and its scaling levels define a conformally invariant theory, that we call regime IV and which extends the three previously known regimes for the spin-one model. The new regime IV provides the actual description of the (analytically continued) colouring problem within a much wider range, namely 2 < Q <= 4, although g_2 is only dominant on the smaller range. The corresponding conformal field theory is identified and the exact critical exponents are derived. We discuss their implications for the phase diagram of the antiferromagnetic triangular-lattice Potts model at non-zero temperature. Finally, we relate our results to recent observations in the field of spin-one anyonic chains.

We present results for the exact perimeter generating functions of several new types of column-convex polygons on the square lattice. We shall focus on a particular (surprising?) example which contains a cautionary warning that even after we have found a closed form solution our work may not be done. In the best tradition of Tony this study combines exact solutions and asymptotic analysis.

Quasicrystals are examples of non-periodic structures with long-range orientational order. They are described by so-called regular model sets, i.e., projections of certain subsets of a lattice in higher dimensional space. We argue that the celebrated diffraction formula for regular model sets is equivalent to the Poisson summation formula for the underlying lattice. This is based on joint work with Nicolae Strungaru from Edmonton.

Tony Guttmann has written over 250 beautiful papers in his long and illustrious career, many about counting problems in statistical mechanics, combinatorics and chemistry. Remarkably, only two of his papers have the word "integral" in the title and none deal with virtual Koornwinder integrals. In this talk I will predict this to dramatically change with Tony's next 250 papers, by showing that such integrals are very effective in solving deep counting problems.

We develop a new approach for uniform generation of random graphs with given degrees, and apply it to derive a uniform sampling algorithm for $d$-regular graphs. The algorithm can be implemented such that each graph is generated in expected time $O(nd^3)$, provided that $d=o(\sqrt{n})$. Our result significantly improves the previously best uniform sampler, which works efficiently only when $d=O(n^{1/3})$, with essentially the same running time for the same $d$. We also give a linear-time approximate sampler which, for $d=o(\sqrt{n})$, generates a random $d$-regular graph whose distribution differs from the uniform by $o(1)$ in total variation distance.This is joint work with Jane Gao.