Invariance Principle for a Potts Interface Along a Wall


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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    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.


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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:


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


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 \(e\in A\),

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

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).

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