Tony Guttmann is widely regarded as one of *Journal of Physics A’s* most prolific authors having published over 100 papers with the journal and refereed many more. He is based at the University of Melbourne and is interested broadly in equilibrium statistical mechanics, and more particularly in discrete models of phase transitions.

**Combinatorics of lattice models: a special issue in honour of Tony Guttmann’s 70th birthday**.

The broad scope of the issue reflects the range of contributions that Tony Guttmann and his collaborators have made to the field. Guest editors are Nathan Clisby and Richard Brak, both based in the mathematics department of the University of Melbourne.

The special issue ties in with the recent Australian and New Zealand Association of Mathematical Physics (ANZAMP) annual meeting for which there was an additional satellite conference Guttmann 2015 – 70 and counting, celebrating statistical mechanics, combinatorics and Tony Guttmann’s 70th birthday.

**1. Could you provide us with a brief overview of your career to date? **

I studied physics and mathematics as an undergraduate at The University of Melbourne, and completed an MSc in Physics there at the end of 1966. In 1967 I moved to Sydney and enrolled in a PhD in mathematics at The University of New South Wales, which I completed in September 1969, and shortly thereafter took up a post-doctoral position in the Theoretical Physics Department at University of London King’s College, which at the time was housed in the old Norfolk Hotel.

In 1971 I returned to Australia, taking up a lectureship in mathematics at the University of Newcastle. I remained there for 15 years, moving up through the ranks, and in 1985 was appointed Professor of Mathematics and Dean of the Faculty of Mathematics, the only such Faculty in Australia. In 1987 I left Newcastle and returned to the University of Melbourne, where I have been ever since, as a Professor of Mathematics.

In 2001 a colleague and I were responsible for the creation of the Australian Mathematical Sciences Institute (AMSI), of which I became Interim Director, but in 2002 my colleagues and I were awarded a large grant to establish the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS). I became the Director of MASCOS, a position I still occupy, while AMSI has been well-served by a string of Directors who have all driven important developments in the mathematical sciences in Australia.

For the last 20 years I have also run a state-wide school mathematics competition, which rewards the best and brightest school students based on performances in tests I set, together with two colleagues. I have served a term as President of the Australian Mathematical Society, as a member of the College of Experts of the Australian Research Council, and as Council member of the Australian Academy of Sciences, as well as internal positions as Head of Department and Deputy Dean at the University of Melbourne.

**2. What are you currently working on?**

Many of the problems I work on relate to properties of graphs that can be drawn on a regular lattice. Typically one counts these using computer based algorithms, and trying to write more efficient algorithms occupies some of my time. Then once one has counted as many such graphs as possible, one is interested in the asymptotic behaviour of the coefficients a_{n }which counts the number of such graphs of size* n* I have also spent a lot of time developing techniques to do this as efficiently as possible. Indeed, a technique called *the method of differential approximants *that I developed, together with Geoff Joyce, back at King’s College in 1970, still is the most powerful method in use today for a large class of problems. Very recently I have realised that it is much more powerful than I thought, in that it can be used to *predict* further coefficients with astonishing accuracy, which can then be used in other methods, such as the ratio method, to get a much better estimate of the asymptotics. For example, with 40 terms of the triangular lattice Ising susceptibility, I can predict the next 200 terms with an accuracy of better than 1 part in 1000000000.

I am also interested in *pattern avoiding permutations (PAPs), *which has some intersection with both algebraic combinatorics and theoretical computer science. Recently, with my colleague Andrew Conway, we showed that 1324-PAPs are different from other length-4 PAPs in that their asymptotic behaviour included a sub-exponential term, which was quite unexpected. Having discovered this term, and developed techniques for finding it, we subsequently found it in a number of other problems, including compressed self-avoiding walks, polygons and bridges. We were happy to report on this work in a recent issue of JPhysA.

**3. A recent paper with JPhysA, Compressed self-avoiding walks, bridges and polygons was part of our special issue: Exactly solved models and beyond. Could you summarise your findings here?**

Yes, this is the paper that I referred to in answer to the previous question. Self-avoiding walks are the primary tool to model long-chain polymers in dilute solutions. In recent years there have been a number of experiments measuring the behaviour of such polymers when pulled from a surface. Here we model the situation where they are pushed toward a surface, and find a quite surprising result, in that there is an additional so-called *stretched exponential *term in the asymptotic behaviour of the coefficients. We show how this term arises even in a simpler model of random walks when subject to a similar compressive force, and then go on to give both a theoretical and calculational explanation. We also predict the exact value of the critical exponent of the stretched exponential term, and link this to the fractal dimension of the underlying polymer model.

**4.**

**The upcoming conference and special issue are recognition for the notable contributions you have made to physics and mathematical research. What would you say has been your career highlight / biggest achievement to date?**

The method of differential approximants that I referred to above has probably had the greatest impact, both as a tool for extracting asymptotic estimates, and for finding exact solutions when the underlying function is differentiably finite. Being involved in the development of algorithms for counting many of the graphs defining numerous models in lattice statistics has resulted in exponentially faster algorithms. This in turn has allowed us to to produce greatly extended series, and give very precise estimates of critical parameters. My work on the Ising model susceptibility with several talented colleagues remains a landmark, in which we calculated the nature of the scaling function with incredible precision, by virtue of a polynomial time algorithm for generating the series coefficients.

I was also pleased with a couple of papers that grew out of seminal work by S. Smirnov and H. Duminil-Copin in which we extended an important identity of Smirnov’s away from criticality, and used this to find scaling relations between various critical exponents, and in another calculation to prove the value of the critical adsorption fugacity for SAWs adsorbed onto a surface, as first conjectured by M. Batchelor and C. Yung.

Another highlight is joint work with Christoph Richard and Iwan Jensen in which we were able to predict the scaling function for self-avoiding polygons in terms of both perimeter and area.

I have enjoyed particularly long-standing and fruitful collaborations with Mireille Bousquet-Mélou, Ian Enting, Iwan Jensen and Stu Whittington, and it has been an unalloyed pleasure to work with such talented and engaging colleagues.

But my greatest achievement is the two dozen or more young men and women who I have had the pleasure to work with during the course of their PhD and post-doctoral studies. I have been fortunate in all these interactions, and it has been a great pleasure for me to see all of them move on to successful careers, some in industry, some in academia.

**5. Finally, do you have any advice for young researchers entering the field?**

University life is different now to when I started my career. Universities are closer to businesses now, and there is an unfortunate divide between those seen to be running the universities and the community of scholars which is what a university should be all about. As a consequence, people entering the field have to be not only excellent researchers, but also good teachers, and be willing to carry their share of the administrative load. They will also be subject to sometimes ludicrous metrics to track their performance, and, with students carrying increasing financial burdens, academic staff are increasingly being forced into the role of service providers to customers, rather than the more traditional scholar’s role. But if young researchers go into this with their eyes open, it can still be one of the most rewarding professions, where for much of the time you are paid to think about problems that interest you.

**Combinatorics of lattice models special issue**

The study of the combinatorics of lattice models in statistical mechanics has been an active area of research for over fifty years, and has played a key role in improving our understanding of critical phenomena. The field of lattice combinatorics is as vibrant as ever. For example, in recent years progress has been made in inventing new and interesting models which can be solved exactly, and in developing new enumeration algorithms for models which have not yet been solved, such as self-avoiding walks.

**Further reading **

Self-avoiding walks on the simple cubic lattice – D MacDonald, S Joseph, D L Hunter, L L Moseley, N Jan and A J Guttmann

Vicious walkers and young tableaux I: Without walls – Anthony J Guttmann, Aleksander L Owczarek and Xavier G Viennot

Self-avoiding polygons on the square lattice – Iwan Jensen and Anthony J Guttmann

This work is licensed under a Creative Commons Attribution 3.0 Unported License. Image courtesy of Tony Guttmann, credit for image: Craig Tracy.

Categories: Journal of Physics A: Mathematical and Theoretical