arXiv subjects

Andrew Elvey Price

Publications and source records attributed to Andrew Elvey Price.

25 records · Page 2Linked to original sources

arXiv2019

Eulerian orientations and the six-vertex model on planar map

We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.

math.CO↗

arXiv2018

Counting Planar Eulerian Orientations

Inspired by the paper of Bonichon, Bousquet-Mélou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $const\cdot (1 - μx)/\log(1 - μx),$ where we conjecture that $μ= 4π$ for Eulerian orientations counted by edges and $μ=4\sqrt{3}π$ for 4-valent Eulerian orientations counted by vertices.

math.CO↗

arXiv2017

Numerical studies of Thompson's group F and related groups

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We have also given an improved algorithm for the coefficients of Thompson's group $F,$ giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson's group $F$ using the method from \cite{HHR15} applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson's group $F$ we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics\footnote{ }. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability. We estimate the asymptotics of the cogrowth coefficients of $F$ to be $$ c_n \sim c \cdot μ^n \cdot κ^{n^σ\log^δ{n}} \cdot n^g,$$ where $μ\approx 15,$ $κ\approx 1/e,$ $σ\approx 1/2,$ $δ\approx 1/2,$ and $g \approx -1.$ The growth constant $μ$ must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture \cite{ERvR15, HHR15} that the group is not amenable.

math.GR↗

arXiv2016

The isoperimetric number of the incidence graph of PG(n,q)

Let $Γ_{n,q}$ be the point-hyperplane incidence graph of the projective space $\operatorname{PG}(n,q)$, where $n \ge 2$ is an integer and $q$ a prime power. We determine the order of magnitude of $1-i_V(Γ_{n,q})$, where $i_V(Γ_{n,q})$ is the vertex-isoperimetric number of $Γ_{n,q}$. We also obtain the exact values of $i_V(Γ_{2,q})$ and the related incidence-free number of $Γ_{2,q}$ for $q \le 16$.

math.CO↗

arXiv2016

Permutations sortable by two stacks in series

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further coefficients approximately. Analysing these coefficients by a variety of methods we conclude that the OGF behaves as $$S(z) \sim A (1 - μ\cdot z)^γ,$$ where $μ=12.45 \pm 0.15,$ $γ= 1.5 \pm 0.3,$ and $A \approx 0.02$.

math.CO↗

arXiv2016

Permutations sortable by deques and by two stacks in parallel

Recently Albert and Bousquet-Mélou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We show that the equally long-standing problem of the number of permutations sortable by a double-ended queue (deque) can be simply related to the solution of the same functional equations. Subject to plausible, but unproved, conditions, the radius of convergence of both generating functions is the same. Numerical work confirms this conjecture to 10 significant digits. Further numerical work suggests that the coefficients of the deque generating function behave as $κ_d \cdot μ^n \cdot n^{-3/2},$ where $μ= 8.281402207\ldots,$ while the coefficients of the corresponding tsip generating function behave as $κ_p \cdot μ^n \cdot n^γ$ with $γ\approx -2.473.$ The constants $κ_d$ and $κ_p$ are also estimated. {\em Inter alia,} we study the asymptotics of quarter-plane loops, starting and ending at the origin, with weight $a$ given to north-west and east-south turns. The critical point varies continuously with $a,$ while the corresponding exponent variation is found to be continuous and monotonic for $a > -1/2,$ but discontinuous at $a=-1/2.$

math.CO↗

arXiv2012

Off-critical parafermions and the winding angle distribution of the O($n$) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents $γ_1$ and $γ_{11}$ as well as wedge critical exponents $γ_2(α)$ and $γ_{21}(α)$ and the exponent characterising the winding angle distribution of the O($n$) model in the half-plane, or more generally in a wedge of wedge-angle $α.$ We assume only the existence of these exponents and, for some values of $n,$ the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality $γ_1-γ_{11} \ge 1,$ and its extension $γ_2(α)-γ_{21}(α) \ge 1$ for the edge exponents. We provide conjectured values for all exponents for $n \in [-2,2).$

math-ph↗