Phase transition in random planar diagrams and RNA-type matching

Andrey Y. Lokhov, Olga V. Valba, Mikhail V. Tamm, and Sergei K. Nechaev

Phys. Rev. E 88, 052117 – Published 11 November 2013

Abstract

We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values zero and one. We show that the existence of a perfect planar matching structure is possible only above a certain critical density, $p_{c}$ , of allowed contacts (i.e., of ones). Using a formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, $p_{c}$ , in the thermodynamic limit. This estimation is close to the critical value, $p_{c} \approx 0.379$ , obtained in numerical simulations based on an exact dynamical programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at $T = 0$ occurs at $p_{c}$ .

Received 17 August 2013

DOI:https://doi.org/10.1103/PhysRevE.88.052117

Authors & Affiliations

Andrey Y. Lokhov¹, Olga V. Valba^1,2, Mikhail V. Tamm³, and Sergei K. Nechaev^1,4

¹Université Paris-Sud/CNRS, LPTMS, UMR8626, Bât. 100, 91405 Orsay, France
²Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia
³Physics Department, Moscow State University, 119992, Moscow, Russia
⁴P. N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991, Moscow, Russia

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Vol. 88, Iss. 5 — November 2013

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Images

Figure 1
Examples of imperfect (a) and perfect (b) planar matchings (open dots represent gaps); the gapless representation is shown by a Dyck path (c). The arc is given by an “up” and “down” steps at the same height, shown by arrows $↗$ and $↘$ . The part of the walk between arrows is a Dyck path itself.Reuse & Permissions
Figure 2
Computation of $Z$ and $Z (p)$ : (a) Selection of $L / 4$ nontouching arcs on the set of $L$ points ( $L / 2$ open dots remain unmatched); (b) the same problem reformulated as a partitioning of vertical segments (arcs) between open dots (unmatched points). A certain number of partitions are forbidden by the matrix of contacts $A$ .Reuse & Permissions
Figure 3
(a) The fraction of perfect matchings $η_{L} (p)$ as a function of the density $p$ of ones in the contact matrix $A$ for chain lengths $L = 500, 1000, 2000$ , averaged over 10 000 instances. The dashed line corresponds to the thermodynamic limit $L \to \infty$ , yielding the critical value $p_{c} = 0.379$ . (b) The scaling analysis of curves, corresponding to different chain lengths $L$ . The fitting procedure gives the exponent of the transition width $ν \approx 0.5$ .Reuse & Permissions
Figure 4
Convergence of fraction of links, involved in planar binding, $f_{L}$ to the limiting value $f_{\infty}$ in two regimes, $p > p_{c}$ and $p < p_{c}$ . (a) In the perfect phase, the exponential convergence is demonstrated for $p = 0.38$ and $p = 0.4$ in the semilogarithmic scale. The screening length $ℓ (p)$ diverges as $p$ approaches the critical value $p_{c}$ . (b) In the imperfect phase, the power-law behavior is shown for $p = 0.32$ and $p = 0.34$ in the log-log scale. The exponent $α (p)$ as a function of $p$ takes values between $0.8$ and 1. The data points are averaged over 1000 instances.Reuse & Permissions
Figure 5
The dependence of the coefficient $a (T)$ in (28) on the temperature for $p = 0.15$ , 0.2, 0.25, 0.35, 0.5. For $p > p^{*}$ ( $0.35 < p^{*} < 0.5$ ), the coefficient $a (T)$ seems to follow the $a (T) = 3 T / 2$ law, typical for the molten phase, up to very low temperatures. For $p < p^{*}$ , the $a (T)$ dependence deviates from the high-temperature behavior at some temperature, which we identify as a critical temperature of transition to the glassy phase. The data points are averaged over 10 000 samples.Reuse & Permissions
Figure 6
The phase diagram of the Bernoulli model on the $(T, p)$ plane. The data points correspond to the critical temperature $T_{c}$ of the molten-glass transition for different values $p = 0.15$ , 0.2, 0.25, 0.3, 0.35, 0.5. A four-letter alphabet ( $p = 0.25$ ) is highlighted by a thin dashed line. The critical curve (A–B) separates glassy and molten phases. We conjecture that at zero temperature, the endpoint B, giving $p^{*}$ , coincides with the critical point $p_{c}$ for the perfect-imperfect transition. The thick dashed line (B–C) separates the perfect and imperfect matching cases. The glassy phase lies entirely inside the region, characterized by gaps. Inset: Evidence for the conjecture $p^{*} = p_{c}$ . Study of the pinching free energy $Δ F (L, T)$ at zero temperature. In the limit of large $L$ , the glassy phase is absent for $p > p^{*}$ , characterized by $Δ F (\infty, 0) = 0$ . The point $p^{*}$ can be identified as a crossing point for different $Δ F (L, 0)$ curves, presented here for $L = 1000$ and $L = 2000$ , and its value is found to be very close to $p_{c} = 0.379$ . The data points are averaged over 1000 samples.Reuse & Permissions

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