John Essam | Royal Holloway, University of London (original) (raw)

Papers by John Essam

Research paper thumbnail of Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions

We study the combinatorics of the change of basis of three representations of the stationary stat... more We study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process. Each of the representations considered correspond to a different set of weighted lattice paths which, when summed over, give the stationary state probability distribution. We show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions.

Research paper thumbnail of Analysis of mean cluster size in directed compact percolation near a damp wall

Journal of Statistical Mechanics-Theory and Experiment, 2014

We investigate the behaviour of the mean size of directed compact percolation clusters near a dam... more We investigate the behaviour of the mean size of directed compact percolation clusters near a damp wall in the low-density region, where sites in the bulk are wet (occupied) with probability p while sites on the wall are wet with probability pw. Methods used to find the exact solution for the dry case (pw = 0) and the wet case (pw = 1) turn out to be inadequate for the damp case. Instead we use a series expansion for the pw = 2p case to obtain a second order inhomogeneous differential equation satisfied by the mean size, which exhibits a critical exponent γ = 2, in common with the wet wall result. For the more general case of pw = rp, with r rational, we use a modular arithmetic method of finding ODEs and obtain a fourth order homogeneous ODE satisfied by the series. The ODE is expressed exactly in terms of r. We find that in the damp region 0 < r < 2 the critical exponent γ damp = 1, in common with the dry wall result.

Research paper thumbnail of A combinatorial derivation of the PASEP stationary state

We give a combinatorial derivation and interpretation of the algebra associated with the stationa... more We give a combinatorial derivation and interpretation of the algebra associated with the stationary distribution of the partially asymmetric exclusion process. Our derivation works by constructing a larger Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original chain and that the stationary distribution on this subset is simply related to that of the original chain. This derivation allows us to find expressions for the normalisation using both recurrences and path models. These results exhibit classical combinatorial numbers such as n!, 2n and the Catalan numbers.

Research paper thumbnail of Directed compact percolation: cluster size and hyperscaling

Journal of Physics A: Mathematical and General, 1989

Exact recurrence relations are obtained for the length and size distributions of compact directed... more Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t " gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by

Research paper thumbnail of Directed compact percolation: cluster size and hyperscaling

Journal of Physics A: Mathematical and General, 1989

Exact recurrence relations are obtained for the length and size distributions of compact directed... more Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t " gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by

Research paper thumbnail of REVIEW ARTICLE: Percolation theory

Research paper thumbnail of Low density expansion of the pair correctedness for percolation models

Research paper thumbnail of Mean Length of Finite Clusters in Directed Compact Percolation Near a Damp Wall

Journal of Statistical Physics, 2011

The mean length of finite clusters is derived exactly for the case of directed compact percolatio... more The mean length of finite clusters is derived exactly for the case of directed compact percolation near a damp wall. We find that the result involves elliptic integrals and exhibits similar critical behaviour to the dry wall case.

Research paper thumbnail of Directed compact percolation near a damp wall: mean length and mean number of wall contacts

Journal of Physics A: Mathematical and Theoretical, 2011

Key aspects of the cluster distribution in the case of directed, compact percolation near a damp ... more Key aspects of the cluster distribution in the case of directed, compact percolation near a damp wall are derived as functions of the bulk occupation probability p and the wall occupation probability pw. The mean length of finite clusters and mean number of contacts with the wall are derived exactly, and we find that both results involve elliptic integrals and

Research paper thumbnail of On directed compact percolation near a damp wall

Journal of Physics A: Mathematical and Theoretical, 2009

The percolation probability for directed, compact percolation near a damp wall, which interpolate... more The percolation probability for directed, compact percolation near a damp wall, which interpolates between the previously examined cases, is derived exactly. We find that the critical exponent β = 2 in common with the dry wall, rather than the value previously found in the wet wall and bulk cases. The solution is found via a mapping to a particular model of directed walks. We evaluate the exact generating function for this walk model which is also related to the ASEP model of traffic flow. We compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models.

Research paper thumbnail of Lattice Paths and the Constant Term

Journal of Physics: Conference Series, 2006

We firstly review the constant term method (CTM), illustrating its combinatorial connections and ... more We firstly review the constant term method (CTM), illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems. We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations, the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first posed in 1971. The model stated in 1971 was only solved for a special case-we solve the full model.

Research paper thumbnail of Vicious walkers, flows and directed percolation

Physica A: Statistical Mechanics and its Applications, 1991

It is shown that the problem of m vicious random walkers is equivalent to the enumeration of inte... more It is shown that the problem of m vicious random walkers is equivalent to the enumeration of integer natural flows on a directed square lattice with maximum flow m. An explicit formula for the number of such flows is given as a polynomial in m and is shown to have the same asymptotic form as Fisher's determinantal result. The expected number of such flows on directed bond or site percolation clusters is also a polynomial in m which reduces to the pair connectedness when m = 0. Consequently directed percolation may be seen as a problem of interacting vicious random walkers. Exact results for m= 1 and 2 are given. ge Sm where W°(xo, x, t) is the corresponding number for harmless walkers and S,, is the group of permutations of the m initial coordinates. Harmless walkers move independently so W°(xo,x, t)--fi W°(xjo, Xj, t), (1.2) j=l 0378-4371/91/$03.50 © 1991 -Elsevier Science Publishers B.V. (North-Holland)

Research paper thumbnail of Maximal Fermi walk configurations on the directed square lattice and standard Young tableaux

Journal of Physics A: Mathematical and Theoretical, 2010

ABSTRACT 1 Abstract We consider configurations of n walkers each of which starts at the origin of... more ABSTRACT 1 Abstract We consider configurations of n walkers each of which starts at the origin of a directed square lattice and makes the same number t of steps from node to node along edges of the lattice. Bose walkers are not allowed to cross, but can share edges. Fermi walk configurations must satisfy the additional constraint that no two walkers traverse the same path. Since, for given t, there are only a finite number of t−step paths there is a limit n max on the number of walkers allowed by the Fermi condition. The value of n max is determined for six types of boundary condition. The number of Fermi configurations of n max walkers is also determined using a bijection to standard Young tableaux. In four cases there is no constraint on the endpoints of the walks and the relevant tableaux are shifted.

Research paper thumbnail of Three Attractive Osculating Walkers and a Polymer Collapse Transition

Consider n interacting lock-step walkers in one dimension which start at the points {0, 2, 4,...,... more Consider n interacting lock-step walkers in one dimension which start at the points {0, 2, 4,..., 2(n − 1)} and at each tick of a clock move unit distance to the left or right with the constraint that if two walkers land on the same site their next steps must be in the opposite direction so that crossing is avoided. When two walkers visit and then leave the same site an osculation is said to take place. The space-time paths of these walkers may be taken to represent the configurations of n fully directed polymer chains of length t embedded on a directed square lattice. If a weight l is associated with each of the i osculations the partition function is Z (n) t (l)=;

Research paper thumbnail of New series expansion data for surface and bulk resistivity and conductivity in two-dimensional percolation

Journal of Physics A: Mathematical and General, 1996

ABSTRACT

Research paper thumbnail of Exact enumeration of parallel walks on directed lattices

Journal of Physics A: Mathematical and General, 1993

Mutually avoiding pairs of parallel walks on a number of d-dimensional lattices are mapped onto u... more Mutually avoiding pairs of parallel walks on a number of d-dimensional lattices are mapped onto undirected random walks which return to the origin on a projected (d-1)-dimensional lattice. Generating functions for the number of such pairs of a given length are thus expressed in terms of standard Green's functions. Directed lattices considered include the directed hypercubic and body-centred hypercubic. The generating function for a directed triangular lattice is also obtained. This work is a generalization of known results for the square lattice.

Research paper thumbnail of Determination of Weight Factors in Linked-Cluster Expansions for Lattice Systems

Journal of Mathematical Physics, 1967

Research paper thumbnail of Percolation Processes. II. The Pair Connectedness

Journal of Mathematical Physics, 1971

Research paper thumbnail of Potts models, percolation, and duality

Journal of Mathematical Physics, 1979

Research paper thumbnail of Graph theory and statistical physics

Discrete Mathematics, 1971

Research paper thumbnail of Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions

We study the combinatorics of the change of basis of three representations of the stationary stat... more We study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process. Each of the representations considered correspond to a different set of weighted lattice paths which, when summed over, give the stationary state probability distribution. We show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions.

Research paper thumbnail of Analysis of mean cluster size in directed compact percolation near a damp wall

Journal of Statistical Mechanics-Theory and Experiment, 2014

We investigate the behaviour of the mean size of directed compact percolation clusters near a dam... more We investigate the behaviour of the mean size of directed compact percolation clusters near a damp wall in the low-density region, where sites in the bulk are wet (occupied) with probability p while sites on the wall are wet with probability pw. Methods used to find the exact solution for the dry case (pw = 0) and the wet case (pw = 1) turn out to be inadequate for the damp case. Instead we use a series expansion for the pw = 2p case to obtain a second order inhomogeneous differential equation satisfied by the mean size, which exhibits a critical exponent γ = 2, in common with the wet wall result. For the more general case of pw = rp, with r rational, we use a modular arithmetic method of finding ODEs and obtain a fourth order homogeneous ODE satisfied by the series. The ODE is expressed exactly in terms of r. We find that in the damp region 0 < r < 2 the critical exponent γ damp = 1, in common with the dry wall result.

Research paper thumbnail of A combinatorial derivation of the PASEP stationary state

We give a combinatorial derivation and interpretation of the algebra associated with the stationa... more We give a combinatorial derivation and interpretation of the algebra associated with the stationary distribution of the partially asymmetric exclusion process. Our derivation works by constructing a larger Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original chain and that the stationary distribution on this subset is simply related to that of the original chain. This derivation allows us to find expressions for the normalisation using both recurrences and path models. These results exhibit classical combinatorial numbers such as n!, 2n and the Catalan numbers.

Research paper thumbnail of Directed compact percolation: cluster size and hyperscaling

Journal of Physics A: Mathematical and General, 1989

Exact recurrence relations are obtained for the length and size distributions of compact directed... more Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t " gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by

Research paper thumbnail of Directed compact percolation: cluster size and hyperscaling

Journal of Physics A: Mathematical and General, 1989

Exact recurrence relations are obtained for the length and size distributions of compact directed... more Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t " gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by

Research paper thumbnail of REVIEW ARTICLE: Percolation theory

Research paper thumbnail of Low density expansion of the pair correctedness for percolation models

Research paper thumbnail of Mean Length of Finite Clusters in Directed Compact Percolation Near a Damp Wall

Journal of Statistical Physics, 2011

The mean length of finite clusters is derived exactly for the case of directed compact percolatio... more The mean length of finite clusters is derived exactly for the case of directed compact percolation near a damp wall. We find that the result involves elliptic integrals and exhibits similar critical behaviour to the dry wall case.

Research paper thumbnail of Directed compact percolation near a damp wall: mean length and mean number of wall contacts

Journal of Physics A: Mathematical and Theoretical, 2011

Key aspects of the cluster distribution in the case of directed, compact percolation near a damp ... more Key aspects of the cluster distribution in the case of directed, compact percolation near a damp wall are derived as functions of the bulk occupation probability p and the wall occupation probability pw. The mean length of finite clusters and mean number of contacts with the wall are derived exactly, and we find that both results involve elliptic integrals and

Research paper thumbnail of On directed compact percolation near a damp wall

Journal of Physics A: Mathematical and Theoretical, 2009

The percolation probability for directed, compact percolation near a damp wall, which interpolate... more The percolation probability for directed, compact percolation near a damp wall, which interpolates between the previously examined cases, is derived exactly. We find that the critical exponent β = 2 in common with the dry wall, rather than the value previously found in the wet wall and bulk cases. The solution is found via a mapping to a particular model of directed walks. We evaluate the exact generating function for this walk model which is also related to the ASEP model of traffic flow. We compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models.

Research paper thumbnail of Lattice Paths and the Constant Term

Journal of Physics: Conference Series, 2006

We firstly review the constant term method (CTM), illustrating its combinatorial connections and ... more We firstly review the constant term method (CTM), illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems. We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations, the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first posed in 1971. The model stated in 1971 was only solved for a special case-we solve the full model.

Research paper thumbnail of Vicious walkers, flows and directed percolation

Physica A: Statistical Mechanics and its Applications, 1991

It is shown that the problem of m vicious random walkers is equivalent to the enumeration of inte... more It is shown that the problem of m vicious random walkers is equivalent to the enumeration of integer natural flows on a directed square lattice with maximum flow m. An explicit formula for the number of such flows is given as a polynomial in m and is shown to have the same asymptotic form as Fisher's determinantal result. The expected number of such flows on directed bond or site percolation clusters is also a polynomial in m which reduces to the pair connectedness when m = 0. Consequently directed percolation may be seen as a problem of interacting vicious random walkers. Exact results for m= 1 and 2 are given. ge Sm where W°(xo, x, t) is the corresponding number for harmless walkers and S,, is the group of permutations of the m initial coordinates. Harmless walkers move independently so W°(xo,x, t)--fi W°(xjo, Xj, t), (1.2) j=l 0378-4371/91/$03.50 © 1991 -Elsevier Science Publishers B.V. (North-Holland)

Research paper thumbnail of Maximal Fermi walk configurations on the directed square lattice and standard Young tableaux

Journal of Physics A: Mathematical and Theoretical, 2010

ABSTRACT 1 Abstract We consider configurations of n walkers each of which starts at the origin of... more ABSTRACT 1 Abstract We consider configurations of n walkers each of which starts at the origin of a directed square lattice and makes the same number t of steps from node to node along edges of the lattice. Bose walkers are not allowed to cross, but can share edges. Fermi walk configurations must satisfy the additional constraint that no two walkers traverse the same path. Since, for given t, there are only a finite number of t−step paths there is a limit n max on the number of walkers allowed by the Fermi condition. The value of n max is determined for six types of boundary condition. The number of Fermi configurations of n max walkers is also determined using a bijection to standard Young tableaux. In four cases there is no constraint on the endpoints of the walks and the relevant tableaux are shifted.

Research paper thumbnail of Three Attractive Osculating Walkers and a Polymer Collapse Transition

Consider n interacting lock-step walkers in one dimension which start at the points {0, 2, 4,...,... more Consider n interacting lock-step walkers in one dimension which start at the points {0, 2, 4,..., 2(n − 1)} and at each tick of a clock move unit distance to the left or right with the constraint that if two walkers land on the same site their next steps must be in the opposite direction so that crossing is avoided. When two walkers visit and then leave the same site an osculation is said to take place. The space-time paths of these walkers may be taken to represent the configurations of n fully directed polymer chains of length t embedded on a directed square lattice. If a weight l is associated with each of the i osculations the partition function is Z (n) t (l)=;

Research paper thumbnail of New series expansion data for surface and bulk resistivity and conductivity in two-dimensional percolation

Journal of Physics A: Mathematical and General, 1996

ABSTRACT

Research paper thumbnail of Exact enumeration of parallel walks on directed lattices

Journal of Physics A: Mathematical and General, 1993

Mutually avoiding pairs of parallel walks on a number of d-dimensional lattices are mapped onto u... more Mutually avoiding pairs of parallel walks on a number of d-dimensional lattices are mapped onto undirected random walks which return to the origin on a projected (d-1)-dimensional lattice. Generating functions for the number of such pairs of a given length are thus expressed in terms of standard Green's functions. Directed lattices considered include the directed hypercubic and body-centred hypercubic. The generating function for a directed triangular lattice is also obtained. This work is a generalization of known results for the square lattice.

Research paper thumbnail of Determination of Weight Factors in Linked-Cluster Expansions for Lattice Systems

Journal of Mathematical Physics, 1967

Research paper thumbnail of Percolation Processes. II. The Pair Connectedness

Journal of Mathematical Physics, 1971

Research paper thumbnail of Potts models, percolation, and duality

Journal of Mathematical Physics, 1979

Research paper thumbnail of Graph theory and statistical physics

Discrete Mathematics, 1971