Localization, Big-Jump Regime and the Effect of Disorder for a Class of Generalized Pinning Models

Abstract

One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, other phases may be present. We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered for example in Bar et al. (Phys Rev E 86:061904, 2012). We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the “big-jump” phenomenon for sums of heavy tail random variables under conditioning. We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump.

Appendices

Appendix A: Circular DNA Models

In [6, 7] (to which we refer also for a more complete literature) the problem of modeling circular DNA is considered: circular DNA corresponds notably to the genetic structures called plasmids that are present in cells. Plasmids are also used for genetic manipulations. If a doubled stranded DNA has a circular structure, that is if the strands are not free at their ends but form an ring, then the separation of the two strands – even just locally, global separation may not be possible – generates a conflict with the double helix structure and the physical properties of the DNA polymer. In fact, in a double stranded DNA with free ends, the local separation of the two strands just induces a rotation in the chain. But in the circular case (Fig. 6) this rotation cannot take place: the ring of the two strands has a winding number that can change if the backbone of DNA polymer can absorb, typically with an energetic cost, this (over)twist. Another way of absorbing the winding number is by forming nontrivial spatial structures called supercoils at locations where the two strands are attached (this induces a strain on the base pairs involved, so the relative contact energy changes): everybody has experienced the formation of supercoils when trying to disentangle two ropes, or even just one rope.

Fig. 6

A schematic image of a circular DNA with three loops and four supercoil sections. The thick line represents the segments of DNA on which the two stands are in contact. The thin line represents a single strand portion, and this happens only in loops

The physical phenomena we just described are very complex: the free end case is already of great complexity! Nonetheless the (simple!) Poland–Scheraga model turns out to be a very relevant model for DNA denaturation study in the free end case (see references in [7, 29]). The circular case is tackled in [6, 7]: a model is built from the Poland–Scheraga and the partition function of the model, with our notations, is (1.9). The function $\log \Psi (m,N)$, cf. Def. 1.1, that enters the definition can be seen as a nonlocal energy. Note that the functional dependence in $\Psi$ is only on the length of the polymer and on the number of contacts m. Moreover, with m contacts the total length of the loops is $N-m$.

Let us take a closer look at the models considered in [6, 7]:

• (1) Model with overtwist. This first model is particularly simple:

  $$\begin{aligned} \Psi (m, N)\, =\, \exp \left( -\chi (N-m)^2/m\right) \, , \end{aligned}$$

  (A.1)

  with $\chi >0$. In this case $Q(m, N)\equiv 1$ and $H(\rho )= -\chi (1-\rho )^2/\rho$. We refer to [6, 7] for explanations about the choice of the precise shape of this energy term. Simply put, the smaller the number m of contacts is, the less likely the configuration is. In other words, the opening of loops is penalized. The rationale is that if the loop length increases, then overtwist is produced in the backbone, which is costly from the energetic viewpoint.

• (2) Model with overtwist and supercoils. This is less straightforward and it involves choosing n supercoils among the m contacts. This is done by fair coin flipping: there is no loss of generality in this choice because a bias corresponds simply to an energetic change for supercoil contacts and in the model there is a real parameter w that accounts for that. Here is the choice in [7]:

  $$\begin{aligned} \Psi (m, N)\, =\, \sum _{n=0}^m \frac{1}{2^m} \left( {\begin{array}{c}m\\ n\end{array}}\right) \exp \left( n w - \chi \frac{(N-m-n)^2}{m}\right) \, , \end{aligned}$$

  (A.2)

  where $\chi > 0$ and w are constants. The fair coin structure and the energetic term for supercoils are clear: added to that there is a penalization term that favors $N-n \approx m$. Recall that in the simple overtwist case m close to N is favored. This is simply because n supercoils are formed and the twist that remains has to be absorbed by the portion of DNA, of length $N-n$, which is not in supercoil form.

  The choice (A.2) is in the framework of Definition 1.1 with

  $$\begin{aligned} H(\rho ) = \sup _{\zeta \in [0, \rho ] } \psi (\zeta , \rho ) = \psi (\zeta _0(\rho ), \rho ), \end{aligned}$$

  (A.3)

  where

  $$\begin{aligned} \psi (\zeta , \rho ) := \zeta w -\rho \log 2 - \chi \frac{(1 - \rho -\zeta )^2}{\rho } + \rho \log \rho - \zeta \log ( \zeta ) - (\rho -\zeta ) \log (\rho -\zeta ). \end{aligned}$$

  (A.4)

  Q(m, N) is therefore (implicitly) defined and one can derive the asymptotic behavior for $N\rightarrow \infty$ and m/N asymptotically constant:

  $$\begin{aligned} Q(m,N) \sim q\left( \frac{m}{N}\right) ,\ \text { with }\ q(\rho ) := \sqrt{\frac{\rho }{ \zeta _0( \rho )(\rho - \zeta _0( \rho ))|\partial _\zeta ^2 \psi (\zeta _0( \rho ),\rho )|} }. \end{aligned}$$

  (A.5)

  The function $\psi$ is concave on the convex domain $\{0 \le \zeta \le \rho \le 1\}$. Thus, H is also concave. Moreover H is analytic on (0, 1).

• (3) Model with supercoils. This is the $\chi =\infty$ limit of (A.2). In this limit $N-n=m$ and, since $n\le m$, we have that $m\ge N/2$ and that the opening of a loop must be compensated by at least as many supercoils. Explicitly we obtain

  $$\begin{aligned} \Psi (m,N)\, =\, \frac{1}{2^m} \left( {\begin{array}{c}m\\ N-m\end{array}}\right) e^{ N w -mw} {\mathbf {1}}_{[1/2,1]}(m/N)\, . \end{aligned}$$

  (A.6)

  Since $\Psi (m,N)=0$ for $m<N/2$, this limit case does not fall into the framework of Definition 1.1. A wider set-up that encompasses all models in in [6, 7] can be found in [42, Ch. 4].

We remark that in both examples (1) and (2) $H(0) = -\infty$ and (of course) $H'(0)= \infty$. So, with the convention we have chosen to consider localized both the partly and fully localized cases, the circular DNA models are localized for all values of the parameters and they display a big-jump transition if $\beta =0$ and $\alpha >1$.

Appendix B: On the Strict Convexity of the Disordered Pinning Free Energy

Theorem B.1

Consider the $\Psi \equiv 1$ model, that is the disordered pinning model. For every $\beta \ge 0$ and every $h >h_c(\beta )$ we have $\partial ^2_h \textsc {f}(\beta , h) >0$.

Proof

Let us remark that for $\beta =0$ the result can be established by explicit computations, but the proof that we give here works for the $\beta =0$ case as well. In this proof ${{\mathbf {P}}} _{N, \omega }= {{\mathbf {P}}} _{N, \omega , \beta , h}$ and $\mathrm {Var}_{N, \omega }$ is the variance with respect to ${{\mathbf {P}}} _{N, \omega }$. We know from [37, Proof of Theorem 2.1] that for $h >h_c(\beta )$

$$\begin{aligned} \partial ^2_h \textsc {f}(\beta , h) \, =\, \lim _{N \rightarrow \infty } \frac{1}{N} {{\mathbb {E}}} \, \mathrm {Var}_{N, \omega } \left( \sum _{j=1}^{N-1} \delta _j \right) \,. \end{aligned}$$

(B.1)

We are going to condition on even sites, so let us replace N with 2N and let us denote by ${{\mathcal {F}}} _e$ the $\sigma$-algebra generated by $\delta _j$ with j even: $\mathrm {Var}_{N, \omega , e} (\cdot )$ is going to denote the variance with respect to ${{\mathbf {P}}} _{N, \omega }(\, \cdot \, \vert {{\mathcal {F}}} _e)$. By Jensen’s inequality

$$\begin{aligned} \mathrm {Var}_{2N, \omega } \left( \sum _{j=1}^{2N-1} \delta _j \right) \, \ge \, {{\mathbf {E}}} _{2N, \omega } \left[ \mathrm {Var}_{2N, \omega , e} \left( \sum _{j=1}^{N} \delta _{2j-1} \right) \right] \, . \end{aligned}$$

(B.2)

We know consider the conditional variance on the set $E_\sigma := \{ \tau : \, \delta _{2j} =\sigma _j$ for $j =1,2, \ldots , N-1\}$ for every $\sigma \in \{0,1\}^{\{1,\ldots , N-1\}}$. We set $n(\sigma ):= \sum _{j=1}^{N-1} \sigma _j$ and $\ell _0:=0$ and we define iteratively $\ell _{j+1}= \min \{ \ell > \ell _j: \, \sigma _\ell =1\}$ for $j \le n(\sigma )-1$. We then redifine $\ell _j$ to be $2\ell _j$ and set also $\ell _{n(\sigma )+1}:=2N$. Therefore $\ell _0, \ell _1, \ldots , \ell _{n(\sigma )+1}$ are the $n(\sigma )$ pinned even sites, plus 0 and 2N that are pinned from the start. Note that

$$\begin{aligned} \sum _{j=1}^N \delta _{2j-1} \, =\, \sum _{k=1}^{n(\sigma )+1} \sum _{j=1+\ell _{k-1}/2}^{\ell _k/2} \delta _{2j-1}\, , \end{aligned}$$

(B.3)

and remark that, under ${{\mathbf {P}}} _{N, \omega }(\, \cdot \, \vert {{\mathcal {F}}} _e)(\tau )$ with $\tau \in E_\sigma$, the random variables

$$\begin{aligned} \left( \sum _{j=1+\ell _{k-1}/2}^{\ell _k/2} \delta _{2j-1} \right) _{k=1, \ldots , n(\sigma )+1}\, , \end{aligned}$$

(B.4)

are independent. Therefore on $E_\sigma$

$$\begin{aligned}&\mathrm {Var}_{2N, \omega , e} \left( \sum _{j=1}^{N} \delta _{2j-1} \right) \, =\, \nonumber \\&\quad \sum _{k=1}^{n(\sigma )+1} \mathrm {Var}_{2N, \omega , e} \left( \sum _{j=1+\ell _{k-1}/2}^{\ell _k/2} \delta _{2j-1} \right) \, \ge \, \sum _{\begin{array}{c} k=1, \ldots , n(\sigma )+1 \\ \ell _k -\ell _{k-1}=2 \end{array}} \mathrm {Var}_{2N, \omega , e} \left( \delta _{\ell _k-1} \right) \,. \end{aligned}$$

(B.5)

Since $\delta _{\ell _k-1}$, under the conditional measure we are considering, is just a Bernoulli random variable with parameter (we use the short-cut notation $\omega =\omega _{\ell _k-1}$)

$$\begin{aligned} p( \omega )\, :=\, \frac{K(1)^2 \exp (h + \beta \omega )}{K(1)^2 \exp (h + \beta \omega ) +K(2)}\, , \end{aligned}$$

(B.6)

we see that for k such that $\ell _k-\ell _{k-1}=2$

$$\begin{aligned} \mathrm {Var}_{2N, \omega , e} \left( \delta _{\ell _k-1} \right) \, =\, p\left( \omega _{\ell _k-1} \right) \left( 1- p\left( \omega _{\ell _k-1} \right) \right) \, =: \, \sigma ^2\left( \omega _{\ell _k-1} \right) , \end{aligned}$$

(B.7)

and therefore

$$\begin{aligned} \mathrm {Var}_{2N, \omega , e} \left( \sum _{j=1}^{N} \delta _{2j-1} \right) \, \ge \, {{\mathbf {E}}} _{2N, \omega } \left[ \sum _{k=0}^{N-1} \delta _{2k} \delta _{2k+2}\sigma ^2\left( \omega _{\ell _k-1} \right) \right] \, . \end{aligned}$$

(B.8)

Now we set $\sigma _\star ^2(L):= \inf \{ \sigma ^2(\omega ):\, \vert \omega \vert \le L\}>0$. We remark that $\sigma _\star ^2(L)>0$ for every $L>0$, but in what follows we are forced to work with L such that ${{\mathbb {P}}} (\vert \omega \vert < L)>0$, that is for L above a threshold. With this notation

$$\begin{aligned} \begin{aligned} {{\mathbb {E}}} \mathrm {Var}_{2N, \omega , e} \left( \sum _{j=1}^{N} \delta _{2j-1} \right) \,&\ge \, \sigma _\star ^2(L) {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega } \left[ \sum _{k=0}^{N-1} \delta _{2k}\delta _{2k+2} {\mathbf {1}}_{\vert \omega _{2k+1}\vert \le L} \right] \\&\ge \, \sigma _\star ^2(L) \left( {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega } \left[ \sum _{k=0}^{N-1} \delta _{2k}\delta _{2k+2} \right] - N {{\mathbb {P}}} \left( \vert \omega \vert >L\right) \right) \, , \end{aligned} \end{aligned}$$

(B.9)

and we are left with showing that

$$\begin{aligned} q\, :=\, \liminf _N \frac{1}{N} {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega } \left[ \sum _{k=0}^{N-1} \delta _{2k}\delta _{2k+2} \right] \, >0\, , \end{aligned}$$

(B.10)

because it suffices to choose L so that ${{\mathbb {P}}} \left( \vert \omega \vert >L\right) \le q/2$ to obtain, see (B.1)–(B.2), that $\partial _h^2 \textsc {f}(\beta , h)\ge \sigma _\star ^2(L) q/4>0$.

In order to establish (B.10) we want to show that the quantity under analysis is bounded below by $\lim _NN^{-1} {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega }[\sum _{j=1}^{2N} \delta _j]$, which is equal to $2 \partial \textsc {f}(\beta , h)>0$, times a positive constant. This can be done by explicit estimates, but for sake of conciseness we use that, for any choice of a sequence $(b_N)_{N \in {{\mathbb {N}}} }$ of positive integer numbers satisfying $\lim _N b_N= \infty$ and $\lim _N{b_N}/N=0$, by [37, Theorem 2.2] we have uniformly on $k\in [b_N, N-b_n]\cap {{\mathbb {N}}}$

$$\begin{aligned} \lim _{N \rightarrow \infty } {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega }[\delta _k] \, =\, \partial _h\textsc {f}(\beta , h) \ \ \ \ \text { and } \ \ \ \lim _{N \rightarrow \infty } {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega }[\delta _k\delta _{k+2}]\, =:\, \textsc {l}(\beta ,h)\, , \end{aligned}$$

(B.11)

where the second statement is just the existence of the limit and (B.10) follows once $\textsc {l}(\beta ,h)>0$ is shown. For this we write ${{\mathbf {E}}} _{2N, \omega }[\delta _k\delta _{k+2}]= {{\mathbf {E}}} _{2N, \omega }[\delta _k] {{\mathbf {E}}} _{2N, \omega }[\delta _{k+2}\vert \delta _{k}=1 ]$ and

$$\begin{aligned} {{\mathbf {E}}} _{2N, \omega }[\delta _{k+2}\vert \delta _{k}=1 ] \, =\, \frac{Z_{2, \theta ^{k}\omega , \beta ,h} Z_{N-k-2, \theta ^{k+2}\omega , \beta ,h} }{Z_{N-k, \theta ^{k}\omega , \beta ,h} }\, \ge \, C \exp \left( - \beta \left( \vert \omega _{k+1}\vert + \vert \omega _{k+2}\vert \right) \right) \, , \end{aligned}$$

(B.12)

where the constant $C>0$ depends on h and on $K(\cdot )$: this estimate is a standard surgery procedure ( [29, Ch. 2], full details can be found in [42, Sec. 5.5]) for which one uses notably the regularly varying character of $K(\cdot )$. Therefore

$$\begin{aligned} {{\mathbf {E}}} _{2N, \omega }[\delta _k\delta _{k+2}]\, \ge Ce^{-2 \beta L} {{\mathbf {E}}} _{2N, \omega }[\delta _k] \left( 1-{\mathbf {1}}_{\vert \omega _{k+1}\vert + \vert \omega _{k+2}\vert > L } \right) \, , \end{aligned}$$

(B.13)

and, in turn, we have

$$\begin{aligned} {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega }[\delta _k\delta _{k+2}]\, \ge \, Ce^{-2 \beta L} \left( {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega }[\delta _k]- {{\mathbb {P}}} \left( \vert \omega _{1}\vert > L \right) \right) \, . \end{aligned}$$

(B.14)

It suffices now to choose L so that ${{\mathbb {P}}} \left( \vert \omega _{1}\vert > L \right) \le \partial _h\textsc {f}(\beta , h)/2$ to obtain that, uniformly in k like in (B.11), we have

$$\begin{aligned} \liminf _N {{\mathbb {E}}} {{\mathbf {E}}} _{2N, \omega }\left[ \delta _k\delta _{k+2}\right] \, \ge \, \frac{1}{2} Ce^{-2 \beta L} \partial _h\textsc {f}(\beta , h) \, >\, 0\, , \end{aligned}$$

(B.15)

and we are done. $\square$

We include here the result proved under restrictive conditions in Remark 1.5.

Proposition B.2

For every $\beta \ge 0$ and every h

$$\begin{aligned} \textsc {f}_H(\beta , h)\, \ge \, H(0)\, . \end{aligned}$$

(B.16)

Proof

We can assume $H(0)> -\infty$ and, with $b>0$ and u, v and c(b) like in (1.4) of Definition 1.1, we obtain that

$$\begin{aligned}&Z^\Psi _{N, \omega \beta ,h} \, \ge \, c(b) \exp \left( N\min _ {\rho \in (0, b]} H(\rho ) - bN\right) \nonumber \\&\quad {{\mathbf {E}}} \left[ \exp \left( \beta \sum _{j=1}^N (\beta \omega _j +h) \delta _j\right) ; \, \tau _{\lfloor b N \rfloor } =N, \, \frac{\tau }{N}\cap \left( (0, b)\cup (1-b, 1)\right) = \emptyset \right] , \nonumber \\&\, \end{aligned}$$

(B.17)

a bound that is obtained simply by restricting the partition function to the renewals with $\lfloor b N \rfloor$ contacts and all at distance at least bN from the boundary. Therefore

$$\begin{aligned} \liminf _{N \rightarrow \infty } \frac{1}{N} {{\mathbb {E}}} \log Z^\Psi _{N, \omega \beta ,h} \, \ge \, H(0) + \liminf _{N \rightarrow \infty } \frac{1}{N} {{\mathbb {E}}} \log {{\mathbf {E}}} \left[ \exp \left( \sum _{j=1}^N (\beta \omega _j +h) \delta _j\right) ; \, E_{N, b} \right] \, , \end{aligned}$$

(B.18)

with $E_{N, b}:= \{ \tau _{\lfloor b N \rfloor } =N, \, (\tau /N)\cap \left( (0, b)\cup (1-b, 1)\right) = \emptyset \}$. Using ${{\mathbf {P}}} '(\cdot ):= {{\mathbf {P}}} (\cdot \vert E_{N, b})$ we see that by Jensen’s inequality the quantity of which we take inferior limit in the right-hand side of the last expression is bounded below by

$$\begin{aligned} \frac{1}{N} {{\mathbb {E}}} \left[ \sum _{j=1}^N (\beta \omega _j +h) {{\mathbf {E}}} '[\delta _j] \right] + \frac{1}{N} \log {{\mathbf {P}}} \left( E_{N, b}\right) \, =\, h \frac{ \lfloor bN\rfloor }{N}+ \frac{1}{N} \log {{\mathbf {P}}} \left( E_{N, b}\right) . \end{aligned}$$

(B.19)

Of course the limit of the first term is hb, which can be made arbitrarily small by choosing b small. The remaining term is bounded below, for $N \rightarrow \infty$, by a (negative) quantity that vanishes as $b \searrow 0$ because ${{\mathbf {P}}} (E_{N, b})$ is bounded below by $K(\lceil bN\rceil )^2$ times ${{\mathbf {P}}} ( \tau _{\lfloor b N \rfloor -2}= N- 2\lceil bN\rceil )$, so, by Proposition 3.1, $\liminf _N (1/N) \log {{\mathbf {P}}} (E_{N, b})=0$ for $\alpha >1$ and it vanishes as $b \searrow 0$ for $\alpha \in (0,1]$. $\square$