School of Mathematics and Statistics - Research Publications
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ItemNo Preview AvailablePolymer collapse of a self-avoiding trail model on a two-dimensional inhomogeneous lattice(ELSEVIER, 2022-10-15)
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ItemCritical scaling of lattice polymers confined to a box without endpoint restriction(SPRINGER, 2022-11)Abstract We study 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. This is a natural model of a confined polymer in dilute solution such as polymers in mesoscopic pores. 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 that 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: the density exponent $$\alpha =1/2$$ α = 1 / 2 , the finite size crossover exponent $$1/\nu =4/3$$ 1 / ν = 4 / 3 and the critical partition function exponent $$2-\eta =43/24$$ 2 - η = 43 / 24 . This implies that the theoretical framework relating them to the unconstrained SAW problem is valid.
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ItemExact solution of some quarter plane walks with interacting boundaries(ELECTRONIC JOURNAL OF COMBINATORICS, 2019-09-13)The set of random walks with different step sets (of short steps) in the quarter plane has provided a rich set of models that have profoundly different integrability properties. In particular, 23 of the 79 effectively different models can be shown to have generating functions that are algebraic or differentiably finite. Here we investigate how this integrability may change in those 23 models where in addition to length one also counts the number of sites of the walk touching either the horizontal and/or vertical boundaries of the quarter plane. This is equivalent to introducing interactions with those boundaries in a statistical mechanical context. We are able to solve for the generating function in a number of cases. For example, when counting the total number of boundary sites without differentiating whether they are horizontal or vertical, we can solve the generating function of a generalised Kreweras model. However, in many instances we are not able to solve as the kernel methodology seems to break down when including counts with the boundaries.
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ItemSelf-avoiding walks in a slab with attractive walls(IOP PUBLISHING LTD, 2005-12-16)
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ItemScaling analysis for the adsorption transition in a watermelon network of n directed non-intersecting walks(KLUWER ACADEMIC/PLENUM PUBL, 2001-02)
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ItemA directed walk model of a long chain polymer in a slit with attractive walls(IOP PUBLISHING LTD, 2005-05-20)