Exact solution of weighted partially directed walks crossing a square. (English) Zbl 1538.82031
Summary: We consider partially directed walks crossing a \(L \times L\) square weighted according to their length by a fugacity \(t\). The exact solution of this model is computed in three different ways, depending on whether \(t\) is less than, equal to or greater than 1. In all cases a complete expression for the dominant asymptotic behaviour of the partition function is calculated. The model admits a dilute to dense phase transition, where for \(0 < t < 1\) the partition function scales exponentially in \(L\) whereas for \(t > 1\) the partition function scales exponentially in \(L^2\), and when \(t = 1\) there is an intermediate scaling which is exponential in \(L\log L\). As such we provide an exact solution of a model of the dilute to dense polymeric phase transition in two dimensions.
MSC:
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 82D60 | Statistical mechanics of polymers |
| 82C23 | Exactly solvable dynamic models in time-dependent statistical mechanics |
References:
| [1] | Bauerschmidt, R.; Brydges, D. C.; Slade, G., Critical two-point function of the 4-dimensional weakly self-avoiding walk, Commun. Math. Phys., 338, 169-93 (2015) · Zbl 1320.82031 · doi:10.1007/s00220-015-2353-5 |
| [2] | Bousquet-Mélou, M.; Guttmann, A. J.; Jensen, I., Self-avoiding walks crossing a square, J. Phys. A: Math. Gen., 38, 9159-81 (2005) · Zbl 1078.82009 · doi:10.1088/0305-4470/38/42/001 |
| [3] | Bradly, C. J.; Owczarek, A. L., Critical scaling of lattice polymers confined to a box without endpoint restriction, J. Math. Chem., 60, 1903-20 (2022) · Zbl 1500.82003 · doi:10.1007/s10910-022-01387-y |
| [4] | Brak, R.; Dyke, P.; Lee, J.; Owczarek, A. L.; Prellberg, T.; Rechnitzer, A.; Whittington, S. G., A self-interacting partially directed walk subject to a force, J. Phys. A: Math. Theor., 42 (2009) · Zbl 1162.82011 · doi:10.1088/1751-8113/42/8/085001 |
| [5] | Burkhardt, T. W.; Guim, I., Self-avoiding walks that cross a square, J. Phys. A: Math. Gen., 24, L1221 (1991) · doi:10.1088/0305-4470/24/20/003 |
| [6] | Cardy, J. L., Conformal invariance and surface critical behavior, Nucl. Phys. B, 240, 514-32 (1984) · doi:10.1016/0550-3213(84)90241-4 |
| [7] | Carmona, P.; Pétrélis, N., Interacting partially directed self avoiding walk: scaling limits, Electron. J. Probab., 21, 1-52 (2016) · Zbl 1346.60136 · doi:10.1214/16-EJP4618 |
| [8] | Forgacs, G., Unbinding of semiflexible directed polymers in 1+1 dimensions, J. Phys. A: Math. Theor., 24, L1099 (1991) · doi:10.1088/0305-4470/24/18/006 |
| [9] | Guttmann, A. J.; Jensen, I., Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, J. Phys. A: Math. Theor., 55 (2022) · Zbl 1515.82070 · doi:10.1088/1751-8121/aca3de |
| [10] | Guttmann, A. J.; Jensen, I.; Owczarek, A. L., Self-avoiding walks contained within a square, J. Phys. A: Math. Theor., 55 (2022) · Zbl 1515.82071 · doi:10.1088/1751-8121/ac9439 |
| [11] | Janse van Rensburg, E. J.; Prellberg, T., Forces and pressures in adsorbing partially directed walks, J. Phys. A: Math. Theor., 49 (2016) · Zbl 1342.82067 · doi:10.1088/1751-8113/49/20/205001 |
| [12] | Janse van Rensburg, E. J.; Prellberg, T.; Rechnitzer, A., Partially directed paths in a wedge, J. Comb. Theory A, 115, 623-50 (2008) · Zbl 1160.05003 · doi:10.1016/j.jcta.2007.08.003 |
| [13] | Knuth, D. E., Mathematics and computer science: coping with finiteness, Science, 194, 1235-42 (1976) · Zbl 1225.68001 · doi:10.1126/science.194.4271.1235 |
| [14] | Knuth, D. E., The Art of Computer Programming, Volume 1: Fundamental Algorithms (1997), Reading, MA: Addison-Wesley, Reading, MA · Zbl 0895.68055 |
| [15] | Lam, P-M; Zhen, Y.; Zhou, H.; Zhou, J., Adsorption of externally stretched two-dimensional flexible and semiflexible polymers near an attractive wall, Phys. Rev. E, 79 (2009) · doi:10.1103/PhysRevE.79.061127 |
| [16] | Lawler, G. F.; Schramm, O.; Werner, W., On the scaling limit of planar self-avoiding walk, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Multifractals, Probability and Statistical Mechanics, Applications. In Part the Proceedings of a Special Session Held During the Annual Meeting of the American Mathematical Society, San Diego, CA, USA, January 2002, pp 339-64 (2004), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1069.60089 |
| [17] | Legrand, A.; Pétrélis, N., A sharp asymptotics of the partition function for the collapsed interacting partially directed self-avoiding walk, J. Stat. Phys., 186, 41 (2022) · Zbl 1490.60278 · doi:10.1007/s10955-022-02890-x |
| [18] | Madras, N., Critical behaviour of self-avoiding walks that cross a square, J. Phys. A: Math. Gen., 28, 1535 (1995) · Zbl 0853.60098 · doi:10.1088/0305-4470/28/6/010 |
| [19] | Madras, N.; Slade, G., The Self-Avoiding Walk (1996), Boston, MA: Birkhäuser, Boston, MA · Zbl 0872.60076 |
| [20] | Owczarek, A. L., Exact solution for semi-flexible partially directed walks at an adsorbing wall, J. Stat. Mech. (2009) · doi:10.1088/1742-5468/2009/11/P11002 |
| [21] | Owczarek, A. L., Effect of stiffness on the pulling of an adsorbing polymer from a wall: an exact solution of a partially directed walk model, J. Phys. A: Math. Theor., 43 (2010) · Zbl 1192.82098 · doi:10.1088/1751-8113/43/22/225002 |
| [22] | Owczarek, A. L.; Brak, R.; Rechnitzer, A., Self-avoiding walks in slits and slabs with interactive walls, J. Math. Chem., 45, 113-28 (2009) · Zbl 1169.82314 · doi:10.1007/s10910-008-9371-x |
| [23] | Owczarek, A. L.; Prellberg, T., Exact solution of semi-flexible and super-flexible interacting partially directed walks, J. Stat. Mech. (2007) · doi:10.1088/1742-5468/2007/11/P11010 |
| [24] | Owczarek, A. L.; Prellberg, T.; Brak, R., The tricritical behaviour of self-interacting partially directed walks, J. Stat. Phys., 72, 737-72 (1993) · Zbl 1101.82337 · doi:10.1007/BF01048031 |
| [25] | Vieira, R. S., On the number of roots of self-inversive polynomials on the complex unit circle, Ramanujan J., 42, 363-9 (2017) · Zbl 1422.30013 · doi:10.1007/s11139-016-9804-2 |
| [26] | Whittington, S. G., Self-avoiding walks and polygons confined to a square (2022) · Zbl 1515.82074 |
| [27] | Whittington, S. G.; Guttmann, A. J., Self-avoiding walks which cross a square, J. Phys. A: Math. Gen., 23, 5601-9 (1990) · doi:10.1088/0305-4470/23/23/030 |
| [28] | Zhou, H.; Zhou, J.; Ou-Yang, Z-C; Kumar, S., Collapse transition of two-dimensional flexible and semiflexible polymers, Phys. Rev. Lett., 97 (2006) · doi:10.1103/PhysRevLett.97.158302 |
| [29] | Zhou, Z.; Grimm, J.; Fang, S.; Deng, Y.; Garoni, T. M., Random-length random walks and finite-size scaling in high dimensions, Phys. Rev. Lett., 121 (2018) · doi:10.1103/PhysRevLett.121.185701 |
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