Pinning Model in Random Correlated Environment: Appearance of an Infinite Disorder Regime

Abstract

We study the influence of a correlated disorder on the localization phase transition in the pinning model (Random polymer models, 2007). When correlations are strong enough, an infinite disorder regime arises: large and frequent attractive regions appear in the environment. We present here a pinning model in random binary ($\{-1,1\}$-valued) environment. Defining infinite disorder via the requirement that the probability of the occurrence of a large attractive region is sub-exponential in its size, we prove that it coincides with the fact that the critical point is equal to its minimal possible value, namely $h_c(\beta )=-\beta$. We also stress that in the infinite disorder regime, the phase transition is smoother than in the homogeneous case, whatever the critical exponent of the homogeneous model is: disorder is therefore always relevant. We illustrate these results with the example of an environment based on the sign of a Gaussian correlated sequence, in which we show that the phase transition is of infinite order in presence of infinite disorder. Our results contrast with results known in the literature, in particular in the case of an IID disorder, where the question of the influence of disorder on the critical properties is answered via the so-called Harris criterion, and where a conventional relevance/irrelevance picture holds.

Appendices

Appendix 1: On the Sign of Gaussian Sequences

Let $\mathsf {W}=\{\mathsf {W}_n\}_{n\in \mathbb {N}}$ be a stationary Gaussian process, centered and with unitary variance, and with covariance matrix denoted by $\Upsilon$, the covariance function being $\rho _k = \Upsilon _{i,i+k}$. We also denote $\Upsilon _l$ the covariance matrix of the vector $(\mathsf {W}_1,\ldots ,\mathsf {W}_l)$, which is just a restriction of $\Upsilon$. We recall Assumption (2.11): correlations are non-negative, and power-law decaying,

$$\begin{aligned} \rho _{k}\mathop { \sim }\limits ^{k\rightarrow \infty } c k^{- a}, \quad \text {for some } a>0 \text { and } c>0. \end{aligned}$$

(6.1)

Proof Proof of Proposition 3.2

We recall here a more general lower bound, dealing with the probability for a Gaussian vector to be componentwise larger than some given value.

Lemma 5.8

[3,  Lemma A.3] Under assumption (2.11) with $a<1$, there exist two constants $c,C>0$ such that for every $l\in \mathbb {N}$, one has

$$\begin{aligned} {\mathbb P} \left( \forall i\in \{1,\ldots ,l\},\ \mathsf {W}_i\geqslant A \right) \geqslant c^{-1}\, \exp \left( -c \big ( A \vee C\sqrt{\log l} \big ) ^2 l^{ a} \right) . \end{aligned}$$

(6.2)

Taking $A=0$ in this Lemma gives (3.8).

To simplify notations, we prove the upper bound for the specific sequence $i_k=k$, $k\in \{1,\ldots ,n\}$. The general proof follows the same reasoning. One first observes that for any subset $\{k_1,\ldots ,k_m\}\subset \{1,\ldots ,n\}$, $m\in \mathbb {N}$, one has

$$\begin{aligned} {\mathbb P} \left( \mathsf {W}_i\geqslant 0 \, ;\, \forall i\in \{1,\ldots ,n\}\right) \leqslant {\mathbb P} \left( \mathsf {W}_{k_j}\geqslant 0 \, ;\, \forall j\in \{1,\ldots ,m\}\right) \end{aligned}$$

(6.3)

The idea is that, if the $k_j$’s are sufficiently far one from another, the Gaussian vector $(\mathsf {W}_{k_1},\ldots ,\mathsf {W}_{k_m})$ behaves like an independent one.

Claim 5.9

• If $a<1$, then there exists some $A>0$ such that taking $k_j:=j\lfloor A n^{1- a}\rfloor$ for $j\in \{0,\ldots ,m:= \lceil A^{-1} n^{ a} \rceil \}$, one has some constant $c>0$ such that for all $n\in {\mathbb N}$

  $$\begin{aligned} {\mathbb P} \left( \mathsf {W}_{k_j}\geqslant 0 \, ;\, \forall j\in \{1,\ldots ,m\}\right) \leqslant e^{-c m}. \end{aligned}$$

  (6.4)

• If $a> 1$, then there exists some integer $A>0$ such that taking $k_j:=j A$ for $j\in \{0,\ldots ,m:= \lceil A^{-1}n \rceil \}$, one has some constant $c>0$ such that for all $n\in {\mathbb N}$

  $$\begin{aligned} {\mathbb P} \left( \mathsf {W}_{k_j}\geqslant 0 \, ;\, \forall j\in \{1,\ldots ,m\}\right) \leqslant e^{-c m}. \end{aligned}$$

  (6.5)

This claim, together with (6.3), gives the conclusion. We now prove the claim.

Under ${\mathbb P}$, the vector $(\mathsf {W}_{k_1},\ldots ,\mathsf {W}_{k_m})$ is a Gaussian vector with covariance matrix $\widetilde{\Upsilon }_m$, with $\widetilde{\Upsilon }_{ij} = \Upsilon _{k_i,k_j}=\rho _{|k_j-k_i|}$ for $i,j\in \{1,\ldots ,m\}$. We note $\widetilde{\mathbb P}$ the law of this $m$-dimensional vector. Then if $\widehat{\mathbb P}$ denotes the law of a $m$-dimensional independent standard Gaussian vector ${\mathcal N} (0,\mathrm{Id})$, a change of measure procedure gives thanks to the Cauchy-Schwarz inequality

$$\begin{aligned} \widetilde{\mathbb P} \left( \mathsf {W}_{j}\geqslant 0 \, ;\, \forall j\in \{1,\ldots ,m\}\right) \leqslant \left( \frac{1}{2} \right) ^{m/2} \widehat{\mathbb E} \left[ \left( \frac{\mathrm{d} \widetilde{\mathbb P} }{\mathrm{d} \widehat{\mathbb P} }\right) ^2\right] ^{1/2}. \end{aligned}$$

(6.6)

One has $\frac{\mathrm{d} \widetilde{\mathbb P} }{\mathrm{d} \widehat{\mathbb P} }(X) = (\det \widetilde{\Upsilon }_m)^{-1/2} e^{-\frac{1}{2} \langle (\widetilde{\Upsilon }_m^{-1}-I)X,X\rangle }$ from the definitions of $\widetilde{\mathbb P}$ and $\widehat{\mathbb P}$, so that from a Gaussian computation, one gets

$$\begin{aligned} \widehat{\mathbb E} \left[ \left( \frac{\mathrm{d} \widetilde{\mathbb P} }{\mathrm{d} \widehat{\mathbb P} }\right) ^2\right] ^{1/2} = (\det \widetilde{\Upsilon }_m)^{-1/2}(\det (2 (\widetilde{\Upsilon }_m)^{-1}-I))^{-1/4} = \det (I-V^2)^{-1/4} \end{aligned}$$

(6.7)

where we defined $V:=\widetilde{\Upsilon }_m-I$.

We now estimate $\det (I-V^2)$. Note that the maximal eigenvalue $\widetilde{\lambda }$ of $\widetilde{\Upsilon }_m$ verifies

$$\begin{aligned} \widetilde{\lambda }\leqslant \max _{i\in \{1,\ldots ,n\}} \sum _{j=1}^{n} \widetilde{\Upsilon }_{ij} \leqslant 1+2\sum _{p=1}^{m} \rho _{k_p}. \end{aligned}$$

(6.8)

Then we use the definition of $k_p$ and $m$, and the assumption (2.11) on $(\rho _k)_{k\geqslant 0}$. We get:

• if $a<1$ one has $\widetilde{\lambda }\leqslant 1+c A^{- a} n^{- a(1- a)} m^{1- a} \leqslant 1+ cA^{-1}$,

• if $a>1$ one has $\widetilde{\lambda }\leqslant 1+c A^{- a}$.

In both cases one chooses $A$ large enough so that $\widetilde{\lambda }\leqslant 3/2$. Thus the eigenvalues of $I-V^{2}$ are bounded from below by $1-(\widetilde{\lambda }-1)^2\geqslant 3/4$, so that in the end one has $\det (I-V^2)\geqslant (3/4)^m$. Combining (6.7) and (6.6) one gets

$$\begin{aligned} \bar{\mathbb P} \left( \mathsf {W}_{j}\geqslant 0 \, ;\, \forall j\in \{1,\ldots ,m\}\right) \leqslant \left( \frac{1}{2} \right) ^{m/2} \left( \frac{3}{4} \right) ^{-m/4}\leqslant 3^{-m/4}. \end{aligned}$$

(6.9)

$\square$

Appendix 2: Estimate on the Homogeneous Model

The following Lemma tells that $n\approx \mathtt {F}(h)^{-1}$ is the threshold at which the partition function starts growing exponentially.

Lemma 5.10

We take $\alpha \ne 1$. There exist $\epsilon >0$, and constants $c, c', c''>0$ not depending on $\epsilon$, such that if $0<h\leqslant c'' \epsilon ^{\alpha \wedge 1} \leqslant 1$, then

• one has $Z_{n,h}^\mathrm{pur}\leqslant 1+c (n\mathtt {F}(h))^{\alpha \wedge 1} \leqslant 1+c \epsilon ^{\alpha \wedge 1}$ for every $n\leqslant \epsilon \mathtt {F}(h)^{-1}$;

• one has $Z_{n,h}^\mathrm{pur} \leqslant \exp ( c' \epsilon ^{\alpha \wedge 1 -1} \mathtt {F}(h) n)$ for every $n\geqslant \epsilon \mathtt {F}(h)^{-1}$.

Proof

Because the case $\alpha >1$ is easier (essentially since $\mathtt {F}(h)$ is proportional to $h$), we restrict to the case $\alpha <1$.

Using that ${\mathbf P} (|\tau \cap [0,n]|\geqslant k) \leqslant (1-\bar{\mathrm {K}}(n))^k$, one writes

$$\begin{aligned} Z_{n,h}^\mathrm{pur} = 1+ \sum _{k=1}^{n} (e^{kh} - e^{(k-1)h}) {\mathbf P} (|\tau \cap [0,n]|\geqslant k) \leqslant 1+ \sum _{k=1}^{n_0} h \left( e^h (1- \bar{\mathrm {K}}(n))\right) ^k. \end{aligned}$$

(7.1)

Then, there is some constant $cst.>0$ such that $\bar{\mathrm {K}}(n) \geqslant cst. n^{-\alpha }$. Since $h\in [0,1]$, we have that $e^h (1- \bar{\mathrm {K}}(n)) \leqslant 1 +2h- cst. n^{-\alpha }$. Since $\mathtt {F}(h)$ is of order $h^{1/\alpha }$, we get that, uniformly for $n\leqslant \epsilon \mathtt {F}(h)^{-1}$, one has $n^{-\alpha } \geqslant \epsilon ^{-\alpha } h$. Therefore, if $\epsilon$ is small enough, $e^h (1- \bar{\mathrm {K}}(n)) \leqslant 1 - \frac{1}{2} cst. n^{-\alpha }<1$. From (7.1) one then gets

$$\begin{aligned} Z_{n_0,h}^\mathrm{pur} \leqslant 1+ h \sum _{k=1}^{n} \left( 1- \frac{1}{2} cst. n^{-\alpha } \right) ^k \leqslant 1+ \frac{h}{\frac{1}{2} cst. n^{-\alpha }}, \end{aligned}$$

(7.2)

which gives $Z_{n,h}^\mathrm{pur}\leqslant 1+c (n\mathtt {F}(h))^{\alpha \wedge 1}$, since $\mathtt {F}(h)$ is of order $h^{1/\alpha }$.

For the second point, one uses that for any two integers $n_1,n_2$, decomposing over the first return time after $n_1$, one gets that $e^hZ_{n_1+n_2,h}^\mathrm{pur} \leqslant e^hZ_{ n_1,h}^\mathrm{pur} e^hZ_{ n_2,h}^\mathrm{pur}$. From this, one gets that for any $p\in {\mathbb N}$

$$\begin{aligned} e^h Z_{p n_0,h}^\mathrm{pur} \leqslant \left( e^h Z_{n_0,h}^\mathrm{pur} \right) ^p \leqslant e^{ p \, \frac{c'}{2} \epsilon ^{\alpha } } = e^{ \frac{c'}{2} \epsilon ^{\alpha -1} p\, n_0 \mathtt {F}(h) } \end{aligned}$$

(7.3)

where we used the previous bound on $Z_{n_0,h}^\mathrm{pur}$, the fact that $h\leqslant c'' \epsilon ^{\alpha }$, and the definition of $n_0= \epsilon \mathtt {F}(h)^{-1}$. The general bound for $n\geqslant n_0$ relies on the monotonicity of $Z_{n,h}^\mathrm{pur}$ in $n$. $\square$