×

The self-avoiding walk spanning a strip. (English) Zbl 1226.82024

Summary: We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as \(\beta \rightarrow \beta_c\) of the probability measure on all finite length walks \(\omega \) with the probability of \(\omega \) proportional to \(\beta^{|\omega|}\) where \(|\omega |\) is the number of steps in \(\omega\). (\(\beta_c\) is the reciprocal of the connective constant.) The self-avoiding walk in a strip \(\{z : 0<\operatorname{Im}(z)<y\}\) is defined by considering all self-avoiding walks \(\omega \) in the strip which start at the origin and end somewhere on the top boundary with probability proportional to \(\beta_c^{|\omega|}\). We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height \(y\). This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and \(\text{SLE}_{8/3}\).

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82D60 Statistical mechanics of polymers
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alberts, T., Duminil-Copin, H.: Bridge decomposition of restriction measures. J. Stat. Phys. 140, 467–493 (2010). Archived as arXiv: 0909.0203v1 [math.PR] · Zbl 1197.82055 · doi:10.1007/s10955-010-9999-3
[2] Chayes, J.T., Chayes, L.: Ornstein-Zernike behavior for self-avoiding walks at all noncritical temperatures. Commun. Math. Phys. 105, 221–238 (1986) · doi:10.1007/BF01211100
[3] Clisby, N.: Efficient implementation of the pivot algorithm for self-avoiding walks. J. Stat. Phys. 140, 349–392 (2010). Archived as arXiv: 1005.1444v1 [cond-mat.stat-mech] · Zbl 1197.82057 · doi:10.1007/s10955-010-9994-8
[4] Kennedy, T.: Monte Carlo tests of SLE predictions for 2D self-avoiding walks. Phys. Rev. Lett. 88, 130601 (2002). Archived as arXiv: math/0112246v1 [math.PR] · doi:10.1103/PhysRevLett.88.130601
[5] Kennedy, T.: Conformal invariance and stochastic Loewner evolution predictions for the 2D self-avoiding walk–Monte Carlo tests. J. Stat. Phys. 114, 51–78 (2004). Archived as arXiv: math/0207231v2 [math.PR] · Zbl 1060.82022 · doi:10.1023/B:JOSS.0000003104.35024.f9
[6] Kesten, H.: On the number of self-avoiding walks. J. Math. Phys. 4, 960–969 (1963) · Zbl 0122.36502 · doi:10.1063/1.1704022
[7] Kesten, H.: On the number of self-avoiding walks II. J. Math. Phys. 5, 1128–1137 (1964) · Zbl 0161.37402 · doi:10.1063/1.1704216
[8] Lawler, G.: Conformally Invariant Processes in the Plane. Am. Math. Soc., Providence (2005) · Zbl 1074.60002
[9] Lawler, G.: Schramm-Loewner evolution. In: Sheffield, S., Spencer, T. (eds.) Statistical Mechanics, IAS/Park City Mathematical Series, pp. 231–295. Am. Math. Soc., Providence (2009). Archived as arXiv: 0712.3256v1 [math.PR]
[10] Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16, 917–955 (2003). Archived as arXiv: math/0209343v2 [math.PR] · Zbl 1030.60096 · doi:10.1090/S0894-0347-03-00430-2
[11] Lawler, G., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 2. Proc. Sympos. Pure Math., vol. 72, pp. 339–364. Am. Math. Soc., Providence (2004). Archived as arXiv: math/0204277v2 [math.PR]
[12] Madras, N., Slade, G.: The Self-avoiding Walk. Birkhäuser, Basel (1996) · Zbl 0872.60076
[13] Schramm, O.: A percolation formula. Electron. Commun. Probab. 6, 115–120 (2001). Archived as arXiv: math/0107096v2 [math.PR] · Zbl 1008.60100 · doi:10.1214/ECP.v6-1041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.