Invariance Principle for a Potts Interface Along a Wall

Abstract

We consider nearest-neighbour two-dimensional Potts models, with boundary conditions leading to the presence of an interface along the bottom wall of the box. We show that, after a suitable diffusive scaling, the interface weakly converges to the standard Brownian excursion.

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Notes

  1. 1.

    The stretched \(\sqrt{k}\) rate of decay is used only for minimizing the discussion needed for ruling out \(k > \sqrt{n}\). For the rest of k-s, the usual exponential bounds with decay rate proportional to k hold.

References

  1. 1.

    Abraham, D.B., Reed, P.: Phase separation in the two-dimensional Ising ferromagnet. Phys. Rev. Lett. 33, 377–379 (1974)

    ADS  Article  Google Scholar 

  2. 2.

    Afanasyev, V.I., Geiger, J., Kersting, G., Vatutin, V.A.: Criticality for branching processes in random environment. Ann. Probab. 33(2), 645–673 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for \(q\ge 1\). Probab. Theory Relat. Fields 153(3–4), 511–542 (2012)

    Article  Google Scholar 

  4. 4.

    Bricmont, J., Lebowitz, J.L., Pfister, C.E.: On the local structure of the phase separation line in the two-dimensional Ising system. J. Stat. Phys. 26(2), 313–332 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Campanino, M., Ioffe, D.: Ornstein–Zernike theory for the Bernoulli bond percolation on \({{\mathbb{Z}}}^d\). Ann. Probab. 30(2), 652–682 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Campanino, M., Ioffe, D., Velenik, Y.: Ornstein–Zernike theory for finite range Ising models above \(T_c\). Probab. Theory Relat. Fields 125(3), 305–349 (2003)

    Article  Google Scholar 

  7. 7.

    Campanino, M., Ioffe, D., Velenik, Y.: Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36(4), 1287–1321 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Campanino, M., Ioffe, D., Louidor, O.: Finite connections for supercritical Bernoulli bond percolation in 2D. Markov Process. Relat. Fields 16(2), 225–266 (2010)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    A. De Masi, D. Ioffe, I. Merola, E. Presutti: Metastability and uphill diffusion. Provisional title (in preparation)

  10. 10.

    Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dobrushin, R.: A statistical behaviour of shapes of boundaries of phases. In: Kotecký, R. (ed.) Phase Transitions: Mathematics, Physics, Biology, pp. 60–70. Springer, Berlin (1992)

    Google Scholar 

  12. 12.

    Dobrushin, S., Kotecký, R., Shlosma, S.: Wulff Construction, Volume 104 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1992)

    Google Scholar 

  13. 13.

    Doney, R.A.: The martin boundary and ratio limit theorems for killed random walks. J. Lond. Math. Soc. 58(3), 761–768 (1998)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Duminil-Copin, H., Manolescu, I.: The phase transitions of the planar random-cluster and potts models with \(q\ge 1\) are sharp. Probab. Theory Relat. Fields 164(3), 865–892 (2016)

    Article  Google Scholar 

  15. 15.

    Duraj, J., Wachtel, V.: Invariance principles for random walks in cones. arXiv:1508.07966 (2015)

  16. 16.

    Durrett, R.: On the shape of a random string. Ann. Probab. 7(6), 1014–1027 (1979)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys. 27, 103–136 (1972)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Greenberg, L., Ioffe, D.: On an invariance principle for phase separation lines. Ann. Inst. H. Poincaré Probab. Stat. 41(5), 871–885 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Grimmett, G.: The Random-Cluster Model, Volume 333 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2006)

    Google Scholar 

  20. 20.

    Higuchi, Y.: On some limit theorems related to the phase separation line in the two-dimensional Ising model. Z. Wahrsch. Verw. Gebiete 50(3), 287–315 (1979)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ioffe, D.: Ornstein–Zernike behaviour and analyticity of shapes for self-avoiding walks on \({ Z}^d\). Markov Process. Relat. Fields 4(3), 323–350 (1998)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Ioffe, D.: Multidimensional random polymers: a renewal approach. In Random Walks, Random Fields, and Disordered Systems, Volume 2144 of Lecture Notes in Mathematics, pp. 147–210. Springer, Cham (2015)

  23. 23.

    Ioffe, D., Ott, S., Shlosman S., Velenik, Y.: Critical prewetting in the 2d Ising model (in preparation)

  24. 24.

    Ioffe, D., Shlosman, S., Toninelli, F.L.: Interaction versus entropic repulsion for low temperature Ising polymers. J. Stat. Phys. 158(5), 1007–1050 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Miracle-Sole, S.: Surface tension, step free energy, and facets in the equilibrium crystal. J. Stat. Phys. 79(1–2), 183–214 (1995)

    ADS  Article  Google Scholar 

  26. 26.

    Ott, S., Velenik, Y.: Potts models with a defect line. Commun. Math. Phys. 362(1), 55–106 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Ott, S., Velenik, Y.: Asymptotics of even-even correlations in the Ising model. Probab. Theory Relat. Fields 175(1–2), 309–340 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Skorokhod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl. 1(3), 261–290 (1956)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The research of D. Ioffe was partially supported by Israeli Science Foundation grant 765/18, S. Ott was supported by the Swiss NSF through an early Postdoc.Mobility Grant and Y. Velenik acknowledges support of the Swiss NSF through the NCCR SwissMAP.

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Appendix A: A Monotone Coupling

Appendix A: A Monotone Coupling

For \(\Delta \subset E_{{\mathbb {Z}}^2}\) finite, denote \(\Phi _{a,\Delta }\equiv \Phi _{a,\Delta }^0\) the random-cluster measure in \(\Delta \) with free (0) boundary condition and weights \(e^{\beta }-1\) on edges with both endpoints having nonnegative second coordinate and weight a on the others. In particular, \(\Phi _{0,\Lambda }\) is the random-cluster measure on the half-box \(\Lambda _+\) with free boundary condition and weights \(e^{\beta }-1\).

In this section, we construct a monotone coupling of \(\Phi _{b,\Delta }\) and \(\Phi _{a,\Delta }\) for \(b>a\). The construction follows closely the one used in the proof of [19, Theorem 3.47]. We fix \(\Delta \) and let \(\Delta ^+ = \Delta \cap ({\mathbb {R}}\times {\mathbb {R}}_{\ge 0}) \) and \(\Delta ^- = \Delta \cap ({\mathbb {R}}\times {\mathbb {R}}_{< 0})\); both are seen as the graphs induced by their set of edges, where edges are identified with the corresponding open line segments. For a finite set of edges E, denote by \(\mathrm {o}_{-}(E)\) the number of edges in E with at least one endpoint having negative second coordinate.

Let \(e_1,\dots ,e_{{|}{E_{\Delta }}{|}}\) be an enumeration of the edges of \(\Delta \) and set \(E_i=\{e_1,\dots ,e_i\}\). Let \((U_i)_{i=1}^{{|}{E_{\Delta }}{|}}\) be an i.i.d. family of uniform random variables on [0, 1]. From a realization \(u=(u_i)_{i}\) of \(U=(U_i)_i\), we construct two configurations \(\omega =\omega (u)\) and \(\eta =\eta (u)\) with joint distribution \(\Psi \) as follows:

figureb

Monotonicity of random-cluster measures in their parameters and boundary condition ensures that \(\omega \ge \eta \). Direct computation shows that \(\omega (U)\sim \Phi _{b}\) and \(\eta (U)\sim \Phi _{a}\).

Claim 1

For any \(e_M\in \Delta ^{-}\),

$$\begin{aligned} \Psi (\omega _{e_M}=1,\, \eta _{e_M}=0 \,|\,U_1=u_1,\dots ,U_{M-1}=u_{M-1} ) \ge \frac{b-a}{(b+q)(b+1)} \end{aligned}$$

uniformly over \(u_1,\dots ,u_{M-1}\).

Proof

First, notice that (denoting \(\omega _{E_{M-1}}(u_1,\dots ,u_{M-1})\) the configuration \(\omega \) restricted to \(E_{M-1}\) and similarly for \(\eta \))

$$\begin{aligned}&\Psi (\omega _{e_M}=1,\, \eta _{e_M}=0 \,|\,U_1=u_1,\dots ,U_{M-1}=u_{M-1} )\\&\quad = \Phi _{b}( X_{e_M}=1 \,|\,X_{E_{i-1}} = \omega _{E_{i-1}} ) - \Phi _{a}( X_{e_M}=1 \,|\,X_{E_{i-1}} = \eta _{E_{i-1}} )\\&\quad \ge \Phi _{b}( X_{e_M}=1 \,|\,X_{E_{i-1}} = \omega _{E_{i-1}} ) - \Phi _{a}( X_{e_M}=1 \,|\,X_{E_{i-1}} = \omega _{E_{i-1}} )\\&\quad =\int _{a}^{b} \frac{\mathrm{d}}{\mathrm{d}s} \Phi _{s}( X_{e_M}=1 \,|\,X_{E_{i-1}} = \omega _{E_{i-1}} )\, \mathrm{d}s. \end{aligned}$$

The claim will thus follow once we establish that \(\frac{\mathrm{d}}{\mathrm{d}s} \Phi _{s}( X_{e_M}=1\,|\,X_{E_{i-1}} = \omega _{E_{i-1}} )\ge (b+q)^{-1}(b+1)^{-1}\) for any \(s\le b\). Write \(\Phi _{s}^{*}(\cdot ) = \Phi _{s}( \cdot \,|\,X_{E_{i-1}} = \omega _{E_{i-1}} ) \); this is a random-cluster measure on \( E_{\Lambda }\setminus E_{M-1}\). Let \(X\sim \Phi _{s}^{*}\). Then,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s} \Phi _{s}^{*}( X_{e_M}=1)&= \frac{1}{s} \mathrm {Cov}_s^{*} \bigl ( {|}{\mathrm {o}_{-}(X)}{|},\, X_{e_M} \bigr )\\&= \frac{1}{s} \mathrm {Cov}_s^{*} \bigl ( {|}{\mathrm {o}_{-}(X)}{|} - X_{e_M},\, X_{e_M} \bigr ) + \frac{1}{s} \Phi _{s}^{*}(X_{e_M}=1) \Phi _{s}^{*}(X_{e_M}=0)\\&\ge \frac{1}{s} \frac{s}{s+q} \frac{1}{s+1} \ge \frac{1}{(b+q)(b+1)}, \end{aligned}$$

since \({|}{\mathrm {o}_{-}(X)}{|} - X_{e_M}\) is a nondecreasing function and is thus positively correlated with \(X_{e_M}\) (the remainder follows from finite energy). \(\square \)

As \(\Psi (\omega _{e_M}=1 \,|\,U_1=u_1,\dots ,U_{M-1}=u_{M-1} )\le \frac{b}{1+b}\) (by finite energy), one has

$$\begin{aligned} \Psi (\eta _{e_M}=0 \,|\,\omega _{e_M}=1,\, U_1=u_1,\dots ,U_{M-1}=u_{M-1} ) \ge \frac{b-a}{(b+q)(b+1)} \frac{1+b}{b} = \frac{b-a}{(b+q)b}. \end{aligned}$$

Write \(\epsilon =\epsilon (a,b) = \frac{b-a}{(b+q)b}\). This implies that, for any configuration \(\psi \) and any set \(A\subset E_{\Delta ^-}\) with \(\psi _e=1\) for all \(e\in A\),

$$\begin{aligned} \Psi ( \omega = \psi ,\, \eta _e=1 \forall e\in A) \le (1-\epsilon )^{|A|}\, \Phi _{b}(\psi ). \end{aligned}$$
(78)

Indeed, writing \(D_i=\{\eta _{e_i}=1\}\) if \(e_i\in A\) and \(D_i=\{\eta _{e_i}\in \{0,1\}\}\) otherwise and setting \(D_{E_i}=\bigcap _{j\le i} D_j\), we get

$$\begin{aligned} \frac{\Psi ( \omega = \psi ,\, \eta _e=1 \forall e\in A)}{\Psi (\omega = \psi )}&\le \prod _{i=1}^{{|}{E_{\Lambda }}{|}}\frac{\Psi ( \omega _{e_i} = \psi _{e_i},\, D_i \,|\,\omega _{E_{i-1}} =\psi _{E_{i-1}},\, D_{E_{i-1}})}{\Psi (\omega _{e_i} = \psi _{e_i} \,|\,\omega _{E_{i-1}} =\psi _{E_{i-1}})}\\&\le \prod _{i:\, e_i\in A} \Psi ( \eta _{e_i}=1 \,|\,\omega _{e_i} = 1,\, \omega _{E_{i-1}}=\psi _{E_{i-1}},\, D_{E_{i-1}})\\&\le (1-\epsilon )^{|A|}. \end{aligned}$$

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Ioffe, D., Ott, S., Velenik, Y. et al. Invariance Principle for a Potts Interface Along a Wall. J Stat Phys 180, 832–861 (2020). https://doi.org/10.1007/s10955-020-02546-8

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Keywords

  • Potts model
  • Random cluster model
  • Interface
  • Ornstein–Zernike theory
  • Invariance principle
  • Brownian excursion