Program for A Tour of Combinatorics and Statistical Mechanics: In Memory of Richard Brak

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Date: Monday, 7 February, 2022.

Venue: Evan Williams Theatre in the Peter Hall Building, University of Melbourne. All talks will be shown simultaneously online.

Times are given in AEDT, that is Australian Eastern Daylight Time, which is UTC+11:00.

Click on talk title links to view the corresponding abstract and also links to the talk pdf (if available) as well as a video recording on YouTube.

A YouTube playlist with video recordings of all talks is available here.

A pdf version of the program can be downloaded here as a standard pdf file, and here in booklet form for printing.

8:15am	Welcome (video)
8:30am	Mireille Bousquet-Mélou	A mini-survey on walks in a cone
	I will review some more or less recent results on the enumeration of walks confined to a cone, with an emphasis on algebraicity properties. pdf video
9:00am	Thomas Prellberg	Staircase polygons revisited: an open question by Richard Brak
	Nearly 30 years ago I was working on enumeration of lattice polygons, such as staircase polygons, with respect to area and perimeter. For staircase polygons, there is a generating function solution in terms of q-deformed Bessel functions, which differentiates with respect to perimeter steps in the horizontal and vertical direction. While very elegant, this solution does not reflect that the answer must be symmetric under exchange of horizontal and vertical steps, which bothered Richard. In this talk I revisit this problem and present such a generating function. This involves a foray into q-deformed algebraic functional equations, a subject I first encountered when working with Richard and Tony. pdf video
9:30am	Stuart Whittington	Self-avoiding walks interacting with a surface and subject to a force
	Self-avoiding walks interacting with a surface constitute a useful model of polymer adsorption. We present some rigorous results about self-avoiding walks interacting with a surface and subject to a force that can desorb the walk. We give results about the free energy that establish the form of the phase diagram, and extend this to the cases of uniform stars and uniform combs. This is joint work with Buks van Rensburg. pdf video
10:00am	E. J. Janse van Rensburg	Pulling spiders
	In this talk I give a brief overview of some basic methods for obtaining free energies of cubic lattice models of adsorbing and pulled polymers in terms of the free energies of adsorbing and pulled self-avoiding walks. In addition to pulling adsorbing spiders (these are models of adsorbing star polymers pulled in their central node, and with legs intersecting the adsorbing plane), I will also briefly look at the free energies of other models. The phase diagrams of some of the models will also be discussed. This work was done jointly with Stu Whittington, and with Chris Soteros. pdf video
10:30am	Morning tea
11:00am	Gary Iliev	Localization of semiflexible polymers near a penetrable interface
	In this talk, we will look at directed walk models related to bilateral Dyck paths interacting with two immiscible solvents and a penetrable interface. The paths will also carry a "stiffness" parameter that decorates pairs of collinear steps. By varying the energy and stiffness parameters, we build up a picture of the phase diagram and understanding of the phase behaviour of such polymeric systems. pdf video
11:30am	Chris Soteros	Entanglements in lattice polygons in tubes
	Lattice polygon models are useful for exploring the probability of knotting and linking in polymers. I will review some recent exact and Monte Carlo results regarding how the probability of a given knot or link type scales with polygon size for polygons in tubular subsets of the simple cubic lattice. Exact and theoretical results for the smallest tube that admits non-trivial knots (the infinity x 2 x 1 tube) confirm a conjecture that the entropic critical exponent goes up by one for each prime factor in the knot-decomposition. Transfer-Matrix Monte-Carlo evidence indicates that the same scaling form holds for some other small tube sizes. Much of this work was done in collaboration with Nick Beaton, Jeremy Eng, Kai Ishihara, Puttipong Pongtanapaisan, Rob Scharein, Koya Shimokawa and Mariel Vazquez. pdf video
12:00pm	Andrew Rechnitzer	Trials and tribulations of preserving topology
	Monte Carlo simulations are a big part of understanding the statistical properties of knots. Unfortunately, if one wishes to study curves of fixed knot types then there are very few methods available. This work, with Nick Beaton and Nathan Clisby, is an attempt to adapt existing algorithms to polygons in R3 of fixed topology. It is very much a work in progress, but I will report on our attempts at trying to coerce the (very fast) pivot algorithm to respect topology. pdf video
12:30pm	Lunch
1:30pm	Judy-anne Osborn	Richard Brak as a role model
	Richard Brak was my PhD supervisor, and the best role model I could think of for a young academic, or any academic. I will reflect on some of what I learnt from him, how the world and academia is the richer for his having been in it, and some of what I see as his legacy. pdf video
2:00pm	Tony Guttmann	Spiral walks on the triangular lattice
	One of the problems Richard worked on for his PhD was that of spiral SAWs on the triangular lattice. At around the same time I was working on an extension of this problem in collaboration with George Szekeres, and I will describe these problems. pdf video
2:30pm	Iwan Jensen	Odds and ends about osculating paths
	We present some odds and ends regarding osculating paths with 4 and 5 walkers. Exact enumerations are used to calculate the exact generating function for walkers in watermelon and star configurations and some critical properties are extracted. pdf video
3:00pm	Afternoon tea
3:30pm	Aleks Owczarek	SAW in a box
	We present a study of self-avoiding walks on the square lattice restricted to a square box of side L weighted by a length fugacity without restriction of their end points. The model admits a phase transition between an ‘empty’ phase, where the average length of walks are finite and the density inside large boxes goes to zero, to a ‘dense’ phase, where there is a finite positive density. We prove various bounds on the free energy and develop a scaling theory for the phase transition based on the standard theory for unconstrained polymers. We compare this model to unrestricted walks and walks whose endpoints are fixed at the opposite corners of a box, as well as Hamiltonian walks. We use Monte Carlo simulations to verify predicted values for three key exponents. pdf video
4:00pm	Nicholas Beaton	The powered Catalan numbers
	Much of Richard’s most recent research was dedicated to understanding bijections between different combinatorial structures. In work with Mathilde Bouvel, Veronica Guerrini and Simone Rinaldi we studied a sequence which we called the “powered Catalan numbers” (OEIS A113227) which count, among other things, several different classes of vincular pattern-avoiding permutations, a class of pattern-avoiding inversion sequences, two classes of (partially) directed lattice paths, and a class of labelled trees. I will discuss some of these objects and the various bijections between them. pdf video
4:30pm	Jan de Gier	The one-transit model
	I will discuss joint work with Richard regarding a Dyck path type model with contact energies above and below a horizontal line. The statistical mechanical partition function of this model gives the stationary distribution of the asymmetric exclusion process with open boundaries. pdf video
5:00pm	Closing remarks

8:15am	Welcome (video)

8:30am	Mireille Bousquet-Mélou
A mini-survey on walks in a cone
I will review some more or less recent results on the enumeration of walks confined to a cone, with an emphasis on algebraicity properties. pdf video

9:00am	Thomas Prellberg
Staircase polygons revisited: an open question by Richard Brak
Nearly 30 years ago I was working on enumeration of lattice polygons, such as staircase polygons, with respect to area and perimeter. For staircase polygons, there is a generating function solution in terms of q-deformed Bessel functions, which differentiates with respect to perimeter steps in the horizontal and vertical direction. While very elegant, this solution does not reflect that the answer must be symmetric under exchange of horizontal and vertical steps, which bothered Richard. In this talk I revisit this problem and present such a generating function. This involves a foray into q-deformed algebraic functional equations, a subject I first encountered when working with Richard and Tony. pdf video

9:30am	Stuart Whittington
Self-avoiding walks interacting with a surface and subject to a force
Self-avoiding walks interacting with a surface constitute a useful model of polymer adsorption. We present some rigorous results about self-avoiding walks interacting with a surface and subject to a force that can desorb the walk. We give results about the free energy that establish the form of the phase diagram, and extend this to the cases of uniform stars and uniform combs. This is joint work with Buks van Rensburg. pdf video

10:00am	E. J. Janse van Rensburg
Pulling spiders
In this talk I give a brief overview of some basic methods for obtaining free energies of cubic lattice models of adsorbing and pulled polymers in terms of the free energies of adsorbing and pulled self-avoiding walks. In addition to pulling adsorbing spiders (these are models of adsorbing star polymers pulled in their central node, and with legs intersecting the adsorbing plane), I will also briefly look at the free energies of other models. The phase diagrams of some of the models will also be discussed. This work was done jointly with Stu Whittington, and with Chris Soteros. pdf video

10:30am	Morning tea

11:00am	Gary Iliev
Localization of semiflexible polymers near a penetrable interface
In this talk, we will look at directed walk models related to bilateral Dyck paths interacting with two immiscible solvents and a penetrable interface. The paths will also carry a "stiffness" parameter that decorates pairs of collinear steps. By varying the energy and stiffness parameters, we build up a picture of the phase diagram and understanding of the phase behaviour of such polymeric systems. pdf video

11:30am	Chris Soteros
Entanglements in lattice polygons in tubes
Lattice polygon models are useful for exploring the probability of knotting and linking in polymers. I will review some recent exact and Monte Carlo results regarding how the probability of a given knot or link type scales with polygon size for polygons in tubular subsets of the simple cubic lattice. Exact and theoretical results for the smallest tube that admits non-trivial knots (the infinity x 2 x 1 tube) confirm a conjecture that the entropic critical exponent goes up by one for each prime factor in the knot-decomposition. Transfer-Matrix Monte-Carlo evidence indicates that the same scaling form holds for some other small tube sizes. Much of this work was done in collaboration with Nick Beaton, Jeremy Eng, Kai Ishihara, Puttipong Pongtanapaisan, Rob Scharein, Koya Shimokawa and Mariel Vazquez. pdf video

12:00pm	Andrew Rechnitzer
Trials and tribulations of preserving topology
Monte Carlo simulations are a big part of understanding the statistical properties of knots. Unfortunately, if one wishes to study curves of fixed knot types then there are very few methods available. This work, with Nick Beaton and Nathan Clisby, is an attempt to adapt existing algorithms to polygons in R3 of fixed topology. It is very much a work in progress, but I will report on our attempts at trying to coerce the (very fast) pivot algorithm to respect topology. pdf video

12:30pm	Lunch

1:30pm	Judy-anne Osborn
Richard Brak as a role model
Richard Brak was my PhD supervisor, and the best role model I could think of for a young academic, or any academic. I will reflect on some of what I learnt from him, how the world and academia is the richer for his having been in it, and some of what I see as his legacy. pdf video

2:00pm	Tony Guttmann
Spiral walks on the triangular lattice
One of the problems Richard worked on for his PhD was that of spiral SAWs on the triangular lattice. At around the same time I was working on an extension of this problem in collaboration with George Szekeres, and I will describe these problems. pdf video

2:30pm	Iwan Jensen
Odds and ends about osculating paths
We present some odds and ends regarding osculating paths with 4 and 5 walkers. Exact enumerations are used to calculate the exact generating function for walkers in watermelon and star configurations and some critical properties are extracted. pdf video

3:00pm	Afternoon tea

3:30pm	Aleks Owczarek
SAW in a box
We present a study of self-avoiding walks on the square lattice restricted to a square box of side L weighted by a length fugacity without restriction of their end points. The model admits a phase transition between an ‘empty’ phase, where the average length of walks are finite and the density inside large boxes goes to zero, to a ‘dense’ phase, where there is a finite positive density. We prove various bounds on the free energy and develop a scaling theory for the phase transition based on the standard theory for unconstrained polymers. We compare this model to unrestricted walks and walks whose endpoints are fixed at the opposite corners of a box, as well as Hamiltonian walks. We use Monte Carlo simulations to verify predicted values for three key exponents. pdf video

4:00pm	Nicholas Beaton
The powered Catalan numbers
Much of Richard’s most recent research was dedicated to understanding bijections between different combinatorial structures. In work with Mathilde Bouvel, Veronica Guerrini and Simone Rinaldi we studied a sequence which we called the “powered Catalan numbers” (OEIS A113227) which count, among other things, several different classes of vincular pattern-avoiding permutations, a class of pattern-avoiding inversion sequences, two classes of (partially) directed lattice paths, and a class of labelled trees. I will discuss some of these objects and the various bijections between them. pdf video

4:30pm	Jan de Gier
The one-transit model
I will discuss joint work with Richard regarding a Dyck path type model with contact energies above and below a horizontal line. The statistical mechanical partition function of this model gives the stationary distribution of the asymmetric exclusion process with open boundaries. pdf video

5:00pm	Closing remarks