La marche auto-évitante

Beffara, Vincent

Journées mathématiques X-UPS, Arbres et marches aléatoires (2016), pp. 103-130.

Ce texte présente quelques propriétés de la marche auto-évitante uniforme sur un réseau, ainsi qu’une preuve complète du résultat de Duminil-Copin et Smirnov calculant la constante de connectivité du réseau hexagonal.

[1] Bauerschmidt, Roland; Duminil-Copin, Hugo; Goodman, Jesse; Slade, Gordon Lectures on self-avoiding walks, Probability and statistical physics in two and more dimensions (Clay Math. Proc.), Volume 15, American Mathematical Society, Providence, RI, 2012, pp. 395-467 | MR | Zbl

[2] Bolthausen, Erwin; van der Hofstad, Remco; Kozma, Gady Lace expansion for dummies, Ann. Inst. Henri Poincaré Probab. Stat., Volume 54 (2018) no. 1, pp. 141-153 | DOI | MR | Zbl

[3] Bolthausen, Erwin; Ritzmann, Christine A central limit theorem for convolution equations and weakly self-avoiding walks, 2001 | arXiv

[4] Brydges, David C.; Spencer, Thomas Self-avoiding walk in 5 or more dimensions, Comm. Math. Phys., Volume 97 (1985) no. 1-2, pp. 125-148 | DOI | MR | Zbl

[5] Duminil-Copin, Hugo; Glazman, Alexander; Hammond, Alan; Manolescu, Ioan On the probability that self-avoiding walk ends at a given point, Ann. Probability, Volume 44 (2016) no. 2, pp. 955-983 | DOI | MR | Zbl

[6] Duminil-Copin, Hugo; Hammond, Alan Self-avoiding walk is sub-ballistic, Comm. Math. Phys., Volume 324 (2013) no. 2, pp. 401-423 | DOI | MR | Zbl

[7] Duminil-Copin, Hugo; Smirnov, Stanislav The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$, Ann. of Math. (2), Volume 175 (2010) no. 3, pp. 1653-1665 | DOI | MR | Zbl

[8] Hammersley, J. M.; Welsh, D. J. A. Further results on the rate of convergence to the connective constant of the hypercubical lattice, Q. J. Math., Volume 13 (1962) no. 1, pp. 108-110 | DOI | MR | Zbl

[9] Hara, Takashi; Slade, Gordon Self-avoiding walk in five or more dimensions I. The critical behaviour, Comm. Math. Phys., Volume 147 (1992) no. 1, pp. 101-136 | DOI | MR | Zbl

[10] Hardy, G. H.; Ramanujan, S. Asymptotic formula for the distribution of integers of various types, Proc. London Math. Soc., Volume 16 (1916), pp. 112-132 | DOI | Zbl

[11] Jensen, Iwan Enumeration of self-avoiding walks on the square lattice, J. Phys. A, Volume 37 (2004) no. 21, pp. 5503-5524 | DOI | MR | Zbl

[12] Kesten, Harry On the number of self-avoiding walks, J. Math. Phys., Volume 4 (1963) no. 7, p. 960 | DOI | MR | Zbl

[13] Kesten, Harry On the number of self-avoiding walks. II, J. Math. Phys., Volume 5 (1964) no. 8, p. 1128 | DOI | MR | Zbl

[14] Lawler, Gregory F; Schramm, Oded; Werner, Wendelin On the scaling limit of planar self-avoiding walk, Fractal geometry and applications : a jubilee of Benoît Mandelbrot, Part 2 (Proc. Sympos. Pure Math.), Volume 72, American Mathematical Society, Providence, RI, 2004, pp. 339-364 | DOI | MR | Zbl

[15] Madras, Neal; Slade, Gordon The self-avoiding walk, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2013 (Reprint of the 1993 original) | DOI | MR

[16] Nienhuis, Bernard Exact critical point and critical exponents of $O\left(n\right)$ models in two dimensions, Phys. Rev. Lett., Volume 49 (1982) no. 15, pp. 1062-1065 | DOI | MR

[17] Nienhuis, Bernard Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Statist. Phys., Volume 34 (1984) no. 5-6, pp. 731-761 | DOI | MR | Zbl

[18] O’Brien, George L. Monotonicity of the number of self-avoiding walks, J. Statist. Phys., Volume 59 (1990) no. 3-4, pp. 969-979 | DOI | MR | Zbl

[19] Schramm, Oded Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., Volume 118 (1999), pp. 221-288 | DOI | MR | Zbl

[20] Slade, Gordon The diffusion of self-avoiding random walk in high dimensions, Comm. Math. Phys., Volume 110 (1987) no. 4, pp. 661-683 | DOI | MR | Zbl