Electronic Journal of Probability

On the Critical Point of the Random Walk Pinning Model in Dimension d=3

Quentin Berger and Fabio Toninelli

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We consider the Random Walk Pinning Model studied in [Birkner-Sun 2008] and [Birkner-Greven-den Hollander 2008]: this is a random walk $X$ on $\mathbb{Z}^d$, whose law is modified by the exponential of beta times the collision local time up to time $N$ with the (quenched) trajectory $Y$ of another $d$-dimensional random walk. If $\beta$ exceeds a certain critical value $\beta_c$, the two walks stick together for typical $Y$ realizations (localized phase). A natural question is whether the disorder is relevant or not, that is whether the quenched and annealed systems have the same critical behavior. Birkner and Sun proved that $\beta_c$ coincides with the critical point of the annealed Random Walk Pinning Model if the space dimension is $d=1$ or $d=2$, and that it differs from it in dimension $d$ larger or equal to $4$ (for $d$ strictly larger than $4$, the result was proven also in [Birkner-Greven-den Hollander 2008]). Here, we consider the open case of the marginal dimension $d=3$, and we prove non-coincidence of the critical points.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 21, 654-683.

Accepted: 17 May 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 60K37: Processes in random environments

Pinning Models Random Walk Fractional Moment Method Marginal Disorder

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Berger, Quentin; Toninelli, Fabio. On the Critical Point of the Random Walk Pinning Model in Dimension d=3. Electron. J. Probab. 15 (2010), paper no. 21, 654--683. doi:10.1214/EJP.v15-761. https://projecteuclid.org/euclid.ejp/1464819806

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  • K.S. Alexander and N. Zygouras, Quenched and annealed critical points in polymer pinning models, Comm. Math. Phys. 291 (2009), 659–689.
  • M. Birkner, A. Greven and F. den Hollander, Quenched large deviation principle for words in a letter sequence, Probab. Theory Rel. Fields, to appear, arXiv: 0807.2611v1 [math.PR],
  • M. Birkner and R. Sun, Annealed vs Quenched critical points for a random walk pinning model, Ann. Inst. Henri Poincare' – Probab. Stat., to appear, arXiv: 0807.2752v1 [math.PR],
  • M. Birkner and R. Sun, Disorder relevance for the random walk pinning model in $d=3$, arXiv:0912.1663,
  • B. Davis and D. McDonald, An Elementary Proof of the Local Limit Theorem, J. Theoret. Probab. 8(3) (1995), 693–701.
  • B. Derrida, G. Giacomin, H. Lacoin and F.L. Toninelli, Fractional moment bounds and disorder relevance for pinning models, Comm. Math. Phys. 287 (2009), 867–887.
  • R.A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Relat. Fields 107 (1997), 451–465.
  • G. Giacomin, Random Polymer Models, Imperial College Press, London, 2007
  • G. Giacomin, H. Lacoin and F.L. Toninelli, Marginal relevance of disorder for pinning models, Comm. Pure Appl. Math. 63 (2010), 233–265.
  • G. Giacomin, H. Lacoin and F.L. Toninelli, Disorder relevance at marginality and critical point shift, Ann. Inst. Henri Poincare' – Probab. Stat., to appear, arXiv:0906.1942v1,
  • R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985.
  • H. Lacoin, New bounds for the free energy of directed polymer in dimension 1+1 and 1+2, Comm. Math. Phys. 294 (2010), 471–503.
  • G.F. Lawler, Intersections of random walks, Probability and its Applications. Birkhaeuser, Boston, MA, 1991.
  • F. L. Toninelli, Coarse graining, fractional moments and the critical slope of random copolymers, Electron. Journal Probab. 14 (2009), 531–547.
  • A. Yilmaz, O. Zeitouni, Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three, Commun. Math. Phys., to appear, arXiv:0910.1169,