Watermelon configurations with wall interaction: exact and asymptotic results

We perform an exact and asymptotic analysis of the model of n vicious walkers interacting with a wall via contact potentials, a model introduced by Brak, Essam and Owczarek. More specifically, we study the partition function of watermelon configurations which start on the wall, but may end at arbitrary height, and their mean number of contacts with the wall. We improve and extend the earlier (partially nonrigorous) results by Brak, Essam and Owczarek, providing new exact results, and more precise and more general asymptotic results, in particular full asymptotic expansions for the partition function and the mean number of contacts. Furthermore, we relate this circle of problems to earlier results in the combinatorial and statistical literature.

[1]  Christian Krattenthaler.  An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula , 1998, Discret. Math. Theor. Comput. Sci..

[2]  George D. Birkhoff,et al.  Formal theory of irregular linear difference equations , 1930 .

[3]  Anthony J. Guttmann,et al.  Vicious walkers, friendly walkers and Young tableaux: II. With a wall , 2000 .

[4]  Bhattacharjee,et al.  Reunion and survival of interacting walkers. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  J. W. Essam,et al.  Scaling Analysis for the Adsorption Transition in a Watermelon Network of n Directed Non-Intersecting Walks , 2001 .

[6]  J. Essam,et al.  Exact solution of N directed non-intersecting walks interacting with one or two boundaries , 1999 .

[7]  Anthony J. Guttmann,et al.  Vicious walkers and Young tableaux I: without walls , 1998 .

[8]  A. Verma,et al.  Quadratic transformation formulas for basic hypergeometric series , 1993 .

[9]  Christian Krattenthaler,et al.  The major counting of nonintersecting lattice paths and generating functions for tableaux , 1995 .

[10]  G. Shortley,et al.  The Theory of Complex Spectra , 1930 .

[11]  P. Forrester.  Exact results for vicious walker models of domain walls , 1991 .

[12]  J. Essam,et al.  New Results for Directed Vesicles and Chains near an Attractive Wall , 1998 .

[13]  Curtis Greene,et al.  A New Tableau Representation for Supersymmetric Schur Functions , 1994 .

[14]  Kanwar Sen,et al.  Lattice path approach to transient solution of M/M/1 with (0,k) control policy , 1993 .

[15]  Essam,et al.  Vicious walkers and directed polymer networks in general dimensions. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Peter Paule,et al.  A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities , 1995, J. Symb. Comput..

[18]  Vicious walkers in a potential , 2004, cond-mat/0403220.

[19]  M. Katori,et al.  Functional central limit theorems for vicious walkers , 2002, math/0203286.

[20]  I. Gessel,et al.  Binomial Determinants, Paths, and Hook Length Formulae , 1985 .

[21]  Samuel Karlin,et al.  COINCIDENT PROPERTIES OF BIRTH AND DEATH PROCESSES , 1959 .

[22]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[23]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[24]  Hans-Dietrich O. F. Gronau,et al.  Path systems in acyclic directed graphs , 1987 .

[25]  S. G. Mohanty,et al.  Lattice Path Counting and Applications. , 1980 .

[26]  Vicious random walkers and a discretization of Gaussian random matrix ensembles , 2001, cond-mat/0107221.

[27]  D. Zeilberger,et al.  Resurrecting the asymptotics of linear recurrences , 1985 .

[28]  W. N. Bailey,et al.  Generalized hypergeometric series , 1935 .

[29]  Christian Krattenthaler,et al.  A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin , 1995, Electron. J. Comb..

[30]  Scaling limit of vicious walks and two-matrix model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[32]  Ora Engelberg.  On some problems concerning a restricted random walk , 1965 .

[33]  G. Rw.  Decision procedure for indefinite hypergeometric summation , 1978 .

[34]  Michael E. Fisher,et al.  Walks, walls, wetting, and melting , 1984 .

[35]  Robert A. Sulanke,et al.  A determinant for q-counting n-dimensional lattice paths , 1990, Discret. Math..

[36]  George D. Birkhoff,et al.  Analytic theory of singular difference equations , 1933 .

[37]  Doron Zeilberger,et al.  A fast algorithm for proving terminating hypergeometric identities , 1990, Discret. Math..

[38]  Christian Krattenthaler.  Another Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula , 1999, J. Comb. Theory, Ser. A.

[39]  W. Rheinboldt,et al.  Generalized hypergeometric functions , 1968 .

[40]  Mihai Ciucu,et al.  Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole , 2001, J. Comb. Theory, Ser. A.

[41]  J. W. Essam,et al.  Return polynomials for non-intersecting paths above a surface on the directed square lattice , 2001 .

[42]  T. Nagao,et al.  Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov-de Gennes random matrices. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  B. Lindström.  On the Vector Representations of Induced Matroids , 1973 .

[44]  B. Sagan.  The Symmetric Group , 2001 .

[45]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..

[46]  P. Forrester.  Exact solution of the lock step model of vicious walkers , 1990 .

[47]  Peter J. Forrester.  Probability of survival for vicious walkers near a cliff , 1989 .

[48]  Somendra M. Bhattacharjee,et al.  Reunion of vicious walkers: Results from epsilon -expansion , 1993 .

[49]  John C. Slater,et al.  The Theory of Complex Spectra , 1929 .

[50]  Ilse Fischer.  Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre , 2001, J. Comb. Theory, Ser. A.

[51]  Mihai Ciucu,et al.  Enumeration of Lozenge Tilings of Hexagons with Cut-Off Corners , 2002, J. Comb. Theory, Ser. A.

[52]  J. C. Slater.  Quantum Theory of Matter , 1952 .

[53]  J. W. Essam,et al.  Bicoloured Dyck Paths and the Contact Polynomial for n Non-Intersecting Paths in a Half-Plane Lattice , 2003, Electron. J. Comb..

[54]  Guo-Niu Han,et al.  The Andrews Festschrift , 2001 .

[55]  M. Katori,et al.  Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory , 2005, math/0501218.

[56]  D. Arrowsmith,et al.  Vicious walkers, flows and directed percolation , 1991 .

[57]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .