Note on Bolthausen-Deuschel-Zeitouni's paper on the absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger

TL;DR

This paper revisits and corrects the proof of the absence of a wetting transition for a pinned harmonic crystal in three or more dimensions, extending previous results to more general pinning potentials.

Contribution

It provides a corrected and generalized proof for the absence of a wetting transition in the case of square-potential pinning, removing reliance on an incorrect lower bound.

Findings

Confirmed absence of wetting transition for harmonic crystals with square-pinning in dimensions three and higher.

Provided a new proof method that does not depend on the disputed lower bound.

Connected with recent alternative approaches by Giacomin and Lacoin.

Abstract

The article [Bolthausen et al., 2000] provides a proof of the absence of a wetting transition for the discrete Gaussian free field conditioned to stay positive, and undergoing a weak delta-pinning at height 0. The proof is generalized to the case of a square pinning-potential replacing the delta-pinning, but it relies on a lower bound on the probability for the field to stay above the support of the potential, the proof of which appears to be incorrect. We provide a modified proof of the absence of a wetting transition in the square-potential case, which does not require the aforementioned lower bound. An alternative approach is given in a recent paper by Giacomin and Lacoin.

Equations

$$H_A(\varphi )=\frac{1}{8d}\sum _{x,y\in A\cup \partial A:|x-y|=1}({\varphi }_x-{\varphi }_y{)}^2$$

$$P_A(d\varphi )=\frac{1}{Z_A}{e}^{-H_A(\varphi )}d{\varphi }_A{\delta }_0(d{\varphi }_{A^c})$$

$$\underset{N\to \infty }{\mathrm{liminf}}\underset{l_N^{\Delta }}{\inf }\frac{1}{(2N+1{)}^d}\log P_{A_{l_N^{\Delta }}}(X_i\ge t,i\in A_{l_N^{\Delta }})\ge -\frac{d\log \Delta }{{\Delta }^d}+c_1\frac{\log \log \Delta }{{\Delta }^d}-\frac{c_2{e}^{c_{4}t\sqrt{\log \Delta }}}{{\Delta }^d(\log \Delta {)}^{c_3}}$$

$${\hat{\xi }}_N$$

$${\Omega }_A^{+}$$

$$\mathcal{A}$$

$${\tilde{P}}_{N,a,b}(d\varphi )=\frac{1}{{\tilde{Z}}_{N,a,b}}\exp \left(-H(\varphi )+\sum _{x\in {\Lambda }_N}b\text{mathbbm}{1}_{\left[{\varphi }_x\in [0,a]\right]}\right)d{\varphi }_{{\Lambda }_N}{\delta }_0(d{\varphi }_{{\Lambda }_N^c})$$

$${\hat{P}}_{N,a,b}(d\varphi )=\frac{1}{{\hat{Z}}_{N,a,b}}\exp \left(-H(\varphi )+\sum _{x\in {\Lambda }_N}b\text{mathbbm}{1}_{\left[{\varphi }_x\in [-a,a]\right]}\right)d{\varphi }_{{\Lambda }_N}{\delta }_0(d{\varphi }_{{\Lambda }_N^c}).$$

$${\tilde{P}}_{N,a,b}({\tilde{\xi }}_N<ϵN^d|{\Omega }_N^{+})={\hat{P}}_{N,a,b}({\hat{\xi }}_N<ϵN^d|{\Omega }_N^{+})$$

$${\tilde{P}}_{N,a,b}({\tilde{\xi }}_N>{ϵ}_{b,a}{N}^d|{\Omega }_N^{+})\ge 1-\exp (-{\eta }_{b,a}{N}^d).$$

$${\tilde{P}}_{N,a,b}({\tilde{\xi }}_N<ϵN^d|{\Omega }_N^{+})=\frac{{\tilde{P}}_{N,a,b}(\{{\tilde{\xi }}_N<ϵN^d\}\cap {\Omega }_N^{+})}{{\tilde{P}}_{N,a,b}({\Omega }_N^{+})}$$

$$\exp (\sum _{x\in {\Lambda }_N}b\text{mathbbm}{1}_{\left[{\varphi }_x\in [0,a]\right]})=\prod _{x\in {\Lambda }_N}((e^b-1)\text{mathbbm}{1}_{\left[{\varphi }_x\in [0,a]\right]}+1)$$

$${\tilde{P}}_{N,a,b}({\Omega }_N^{+})$$

$$(\ast \ast )$$

$$={\int }_{[0,a{]}^A}{P}_{A^c}(\varphi +\tilde{\psi }\ge 0on{A}^c)g(\psi )d\psi$$

$$\ge P_{A^c}({\Omega }_{A^c}^{+})$$

$$(\ast )$$

$$=\prod _{i=1}^{|A|}{P}_N({\varphi }_{x_i}\in [0,a]|{\varphi }_{x_{i+1}},\dots ,{\varphi }_{x_{|A|}}\in [0,a])$$

$$=\prod _{i=1}^{|A|}{\int }_{[0,a{]}^{A_i}}{P}_N({\varphi }_{x_i}\in [0,a]|\varphi =\psi on{A}_i)g_i(\psi )d\psi$$

$$(\ast )$$

$$=\prod _{i=1}^{|A|}{\int }_{[0,a{]}^{A_i}}{P}_{A_i^c}({\varphi }_{x_i}+{\tilde{\psi }}_{x_i}\in [0,a])g_i(\psi )d\psi$$

$$\ge \prod _{i=1}^{|A|}{P}_{A_i^c}({\varphi }_{x_i}\in [0,a])$$

$$\ge [c(1/2\wedge a){]}^{|A|}$$

$${\tilde{P}}_{N,a,b}({\Omega }_N^{+})$$

$${\tilde{P}}_{N,a,b}(\{{\tilde{\xi }}_N<ϵN^d\}\cap {\Omega }_N^{+})$$

$$\le \frac{Z_N}{{\tilde{Z}}_{N,a,b}}♯\{A:|A|<ϵN^d\}\exp (JϵN^d)$$

$${\tilde{P}}_{N,a,b}({\tilde{\xi }}_N<ϵN^d|{\Omega }_N^{+})\le \frac{\exp (JϵN^d)♯\{A:|A|<ϵN^d\}}{{\sum }_{A\subset {\Lambda }_N}\exp (J^{\prime }|A|)P_{A^c}({\Omega }_{A^c}^{+})}$$

$$\frac{1}{N^d}\log {\tilde{P}}_{N,a,b}({\tilde{\xi }}_N<ϵN^d|{\Omega }_N^{+})\le$$

$$-\frac{1}{N^d}\log \sum _{A\subset {\Lambda }_N}\exp (J^{\prime }|A|)P_{A^c}({\Omega }_{A^c}^{+})$$

$$\text{eqref}prop3$$

$$\le -\frac{1}{N^d}\left({\left(\frac{N}{\Delta }\right)}^d[(d\log \Delta +c_0)+J^{\prime }-d\log \Delta +c_1\log \log \Delta ]\right)$$

$$\displaystyle=-\frac{J^{\prime}+c_{0}+c_{1}\log\log\Delta}{\Delta^{d}}<0\mbox{ for $\Delta=\Delta(J^{\prime})$ large enough.}$$

Full text

1. Definitions and notations

We keep the notations of [1] except for the field which we call $\phi$ instead of $X$. Let $A$ be a finite subset of $\mathbb{Z}^{d}$, let $\phi=(\phi_{x})_{x\in\mathbb{Z}^{d}}\in\mathbb{R}^{\mathbb{Z}^{d}}$ and the Hamiltonian defined as

[TABLE]

where $\partial A$ is the outer boundary of $A$. The following probability measure on $\mathbb{R}^{A}$ defines the discrete Gaussian free field on $A$ (with zero boundary condition) :

[TABLE]

where $d\phi_{A}=\prod_{x\in A}d\phi_{x}$ and $\delta_{0}$ is the Dirac mass at 0. The partition function $Z_{A}$ is the normalization $Z_{A}=\int_{\mathbb{R}^{A}}\exp(-H(\phi_{A}))d\phi_{A}$. We will also need the following definition of a set $A$ being $\Delta$-sparse (morally meaning that it has only one pinned point per cell of side-length $\Delta$), which we reproduce from [1, page 1215] :

Definition 1**.**

Let $N\in\mathbb{Z}$, $\Delta>0$, $\Lambda_{N}=\{-\lfloor N\rfloor/2,\ldots,\lfloor N\rfloor/2\}^{d}$ and let $l_{N}^{\Delta}=\{z_{i}\}_{i=1}^{|l_{N}^{\Delta}|}$ denote a finite collection of points $z_{i}\in\Lambda_{N}$ such that for each $y\in\Lambda_{N}\cap\Delta\mathbb{Z}^{d}$ there is exacly one $z\in l_{N}^{\Delta}$ such that $|z-y|<\Delta/10$. Let $A_{l_{N}^{\Delta}}=\Lambda_{N}\backslash l_{N}^{\Delta}$.

2. Lower bound on the probability of the hard wall condition

The proof of [1, Theorem 6] relies on [1, Proposition 3]. Unfortunately, the proof provided in the paper, when applied with $t>0$ provides a lower bound which is a little bit weaker than what is claimed, namely

Proposition 2**.**

*Correction of [1, Proposition 3] :

Assume $d\geq 3$ and let $t\geq 0$. Then there exist three constants $c_{1},c_{2},c_{3}>0$ depending on $t$, and $c_{4}>0$ independent of $t$, such that, for all $\Delta$ integer large enough*

[TABLE]

The statement of [1, Proposition 3] only contains the first two terms. The dependence in $t$ vanishes between equations $(2.4)$ and $(2.5)$ in [1]. Note that for $t=0$ the third term is irrelevant and the bound coincides with the one stated in the paper.

3. Proof of the absence of a wetting transition in the square-potential case

Let us introduce the following notations

[TABLE]

and the following probability measure with square-potential pinning :

[TABLE]

in contrast with the measure used in [1] :

[TABLE]

Observe that

[TABLE]

Theorem 3**.**

*(Absence of wetting transition, [1, Theorem 6])

Assume $d\geq 3$ and let $a,b>0$ be arbitrary. Then there exists $\epsilon_{b,a},\eta_{b,a}>0$ such that*

[TABLE]

provided $N$ is large enough.

Proof.

Let us compute the probability of the complementary event and provide bounds on the numerator and the denominator corresponding to the conditional probability :

[TABLE]

3.1. Lower bound on the denominator

Writing

[TABLE]

and using the FKG inequality, we get

[TABLE]

Let us first bound the term $(**)$:

[TABLE]

for some density function $g$. Let $\tilde{\psi}$ be the harmonic extension of $\psi$ to $\Lambda_{N}\backslash A$. Since $\tilde{\psi}\geq 0$, we have

[TABLE]

For the term $(*)$, we write $A=\{x_{1},\ldots,x_{|A|}\},$ and $A_{i}=\{x_{i+1},\ldots,x_{|A|}\},$

[TABLE]

for some density function $g_{i}$. Let $\tilde{\psi}$ be the harmonic extension of $\psi$ to $\Lambda_{N}\backslash A_{i}$, we have

[TABLE]

for some $c=c(d)>0$, since the variance of the free field is bounded in $d\geq 3$. The inequality (17) comes from the fact that $P_{A_{i}^{c}}(\phi_{x_{i}}+\tilde{\psi}_{x_{i}}\in[0,a])\geq P_{A_{i}^{c}}(\phi_{x_{i}}\in[0,a])$ since $\tilde{\psi}_{x_{i}}\in[0,a]$ and $\phi_{x_{i}}$ is a centered Gaussian variable.

Hence,

[TABLE]

with $J^{\prime}=\log(e^{b}-1)+\log c+\log(1/2\wedge a)$.

3.2. Upper bound on the numerator

[TABLE]

with $J=\log(e^{b}-1)+\log(1/2\wedge a)$, where $\sharp X$ denotes the cardinality of the set $X$.

3.3. Upper bound on (5)

[TABLE]

And now we proceed similarily as for the proof with $\delta$-pinning potential:

[TABLE]

The right hand side of (23) can be bounded by $\epsilon(J+1-\log\epsilon)$ as $N$ tends to infinity (by a rough approximation and the Stirling formula), which in turn can be made as close to 0 as we want by choosing $\epsilon=\epsilon(J)$ sufficiently small. See [1].

To bound (24) we use [1, Proposition 3] with $t=0$ which matches to our Proposition 2 :

[TABLE]

where $\Delta$-sparseness corresponds to Definition 1 : a set $A\subset\Lambda_{N}$ is $\Delta$-sparse if it equals $A_{l_{N}^{\Delta}}$, for some set $l_{N}^{\Delta}$.

∎