# Homogeneous Circular DNA

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## Strategy of the proof and structure of the paper

A classical result in the theory of product of random matrices (see e.g. [20, 45]) provides a semi-explicit formula for the Lyapunov exponent, involving an invariant measure for the action of the random matrices on the corresponding projective space. Here, this formula will be obtained in a direct way, and takes the form L() = E[log(1 + 2X)]; (2.1.18) where the law of X is an invariant measure for the random transformation, on [0;+1), x 7! Z 1+x 1+2x : In other words it satisfies X (d) = Z 1 + X 1 + 2X ; (2.1.19).
where Z is independent of X (on the right hand side). Existence and uniqueness of such a random variable X will be justified in Section 2.2, as well as formula (2.1.18). A very useful uniform stochastic dominance of the random variables (X)>0 will also be proved. From that point on, the work will only be based on formula (2.1.18) for the Lyapunov exponent and the fixed point equation (2.1.19). Thanks to the former, the problem will readily boil down to studying X’s moments. That study can be split into two subproblems.
We will know since Section 2.2 which ones of X’s moments are bounded as goes to 0 and which diverge. The two subproblems then are:
– Deriving a regular expansion for X’s bounded moments, involving an error in terms of a divergent moment of X (Sections 2.3 and 2.4);
– Estimating the divergence speed of X’s unbounded moments (Section 2.5).
The former point is addressed in Section 2.3. The analysis is based on a bootstrap procedure, based on recursive uses of the fixed point equation (2.1.19). It gives more and more precise expansions of these moments. Eventually, it will provide the regular expansion (2.1.12) or (2.1.15) with an upper bound on the error R(), involving a divergent moment of X. That work will be generalized in the appendix 2.A, for matrices of size d, with more general entries.
That same strategy, using a bootstrap procedure to obtain a more and more precise estimate of X’s moments, can also provide a lower bound on the error, involving a divergent truncated moment of X: Section 2.4 will be devoted to that analysis.

### Existence and First Properties of the invariant measure X

In this section we prove the existence of the random variables X and derive formula (2.1.18). A first result on X’s moments is also proved: it spells out which moments of X are bounded as goes to 0 and which diverge. We start by introducing an invariant measure of the random matrix M0 ( = 0). It will play a central role to define the random variables X and control their moments. First I need to fix a notation for the stochastic dominance.
Definition 2.2.1. The stochastic dominance will be denoted by 4. Formally, if X and Y are two real-valued random variables, X 4 Y means that P(X > x) 6 P(Y > x) for every x 2 R. Equivalently, there exist two copies ~X and ~ Y , of X and Y respectively, such that ~X 6 ~ Y almost surely. Lemma 2.2.2. Fix a sequence (Zn) of iid copies of Z. The series X0 = X+1 n=1 Z1 Zn.

#### Limiting Behaviour of X’s divergent moments

First note that Theorem 2.1.8 is an immediate consequence of Propositions 2.3.1 and 2.4.1. The goal of this section is to obtain estimates of the error RK(), for which we now have c2(K+1)E[XK+1 1f2X6Bg] 6 RK() 6 C2E[X ]: (2.5.1)
In order to give explicit estimates of RK() in terms of powers of , one needs to understand the limiting behaviour of X’s moments (or truncated moments). The issue was partially addressed by Corollary 2.2.8, which pinpointed the regimes of convergence or divergence of these moments. Namely E[X ] is bounded as goes to 0 if E[Z ] < 1 and diverges if E[Z ] > 1. In the following section we address the issue of the divergence speed when E[Z ] > 1.
The first paragraph, based on renewal theory results, describing the heavy tail of X0, will provide upper bounds for X’s divergent moments. The second paragraph will give lower bounds for these moments under the restriction that Z is bounded.

The Homogeneous Poland-Scheraga Model

In the beginning of the 80s there has been a wave of interest in the physical community for the exactly solvable character of the homogeneous PS model and a number of approaches, motivated both by different physical applications, and by somewhat different methods of solution. In 1984 Michael Fisher  observed that the solvable nature of the model can be seen in full generality in terms of a rather simple computation. In reality, as it has been pointed out in [57, App. A], Fisher’s simple computation was a well-known fact in mathematics from the end of the 40s: the homogeneous PS model is just a rewriting of the basic discrete renewal process. And renewal theory greatly developed in the 50s and 60s, so the full solution of the PS model is just a corollary of known mathematics. Let us present the mathematical formalism that is commonly used for the model. This requires a short introduction on renewal processes.
Definition 3.1.1. A process = (k)k>0 is a discrete renewal process – its law is denoted by P – when 0 = 0, and the increments (k)k>1 = (k􀀀k􀀀1)k>1 are IID random variables taking values in N [ f1g = f1; 2; g [ f1g. We will use the notation K(n) = P(1 = n): by definition, P n2N K(n) + K(1) = 1.
Note that if k = 1 for one k (that is if K(1) > 0), then k = k+1 = : : : = 1. In this case the renewal is called transient or terminating. Instead, if K(1) = 0, then k < 1 for all k, and the renewal is called persistent. The points (k)k>0 will be called pinned points or contact points and the renewal process will often be seen as a subset of the discrete line N[ f1g. If the renewal is terminating, then is a finite set and it contains the point 1, whereas if the renewal is persistent, then is an infinite subset of N. Moreover, let us recall from now the classical fundamental Renewal Theorem formula (see e.g. [41, § XI.1]).

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Beyond the Poland-Scheraga model

Generalized Poland-Scheraga. It is impossible to give here a proper account of the several models that go toward a more realistic, or at least more detailed, modeling of the DNA structure. But one direction, that we have already briefly mentioned, is the one of trying to account for mismatches and slips in the matching of the two strands, or to account for non exact complementary of the two strands, or even possibly for different length of the two strands. The natural generalization of the PS model, proposed in [48, 78] and mathematically studied in [9, 53], accounts for the richness of the possible DNA structures that we just outlined and, very remarkably, keeps the solvable character of the homogeneous version of the model. This new model is called Generalized Poland-Scheraga (GPS) model. We refer to  for a mathematical introduction to this model and we just quickly mention that the solvable nature is once again connected to a renewal structure embodied in the model, but this time the renewal is two dimensional. The GPS model is of interest to us because of its richer phenomenology: we will find it also in the models that we will consider. More precisely, the GPS has a localization/delocalization transition that corresponds to DNA denaturation, but it has also other transitions that happen in the localized regime, that is when the two strands are in their native state. We stress from now that these new transitions have been identified rigorously only in absence of disorder, so, till specified otherwise, we limit ourselves to the non disordered case. What happens in the GPS model if the two strands have different lengths M > N (and if the strands are bound together at both extremities: we make this choice for ease of exposition, see [9, 53] for the general case) is that they can bind together in two different ways:
1. the two strands fully bind together (Figure 3.6) in the sense that essentially all loops have finite length (more precisely, the largest loop is of length O(logN) for N ! 1): this is possible because asymmetric loops are allowed in the GPS model, that is loops that involve a different number of bases for the two stands;
2. the two strands bind together in such a way that essentially all the M 􀀀 N bases that are in excess are absorbed by one large loop (Figure 3.7): again, this is possible because asymmetric loops are allowed. This appearance of a large loop has been presented in then physical literature as a condensation phenomenon. In [9, 53] the transition between localization without a large loop and with a large loop is viewed from Large Deviations perspective as the passage from Cramer Deviations (i.e. deviations that can be captured by an exponential tilt of the measure with which one modifies the expectation of the underlying jumps) to non Cramer Deviations, in which no exponential tilt of the measure can lead to an expectation of the underlying jumps compatible with the constraint that the two chains meet at the extremities. The study of this phenomenon, even beyond Large Deviations and the Cramer versus non Cramer behaviour, is already present in the mathematical literature on random walks and renewal processes and the keyword is: big jump regime.

I On the Lyapunov exponent of random transfer matrices
1 Produits de matrices aléatoires et Modèles d’Ising désordonnés
1.1 Motivation
1.2 Généralités sur l’exposant de Lyapunov
1.2.1 Expression implicite et mesures invariantes
1.2.2 Continuité de l’exposant de Lyapunov
1.2.3 Dérivabilité et Analyticité
1.3 Le Modèle de Derrida–Hilhorst
1.3.1 Développement « régulier » en puissances de 2
1.3.2 Singularité 2
1.4 Modèles en désordre faible
1.4.1 Perturbation faible de la matrice I2
1.4.2 Limite continue du modèle de Derrida–Hilhorst : un modèle exactement soluble
1.A Annexe : Démonstration des Théorèmes 1.4.1 et 1.4.3
1.A.1 Aperçu de la méthode
1.A.2 Trois lemmes sur l’opérateur M2 et démonstration des théorèmes . . 26
1.A.3 Étude de l’opérateur différentiel M2 et preuves des lemmes
1.A.4 Questions ouvertes et Commentaires
2 Regular Expansion for the characteristic exponent of a product of 22 random matrices
2.1 Introduction
2.1.1 General Conjecture and known results
2.1.2 Assumptions and Main Result
2.1.3 Strategy of the proof and structure of the paper
2.2 Existence and First Properties of the invariant measure X
2.3 Regular Expansion (Theorem 2.1.8: upper bound)
2.4 Theorem 2.1.8: lower bound on the error
2.5 Limiting Behaviour of X’s divergent moments
2.5.1 Upper Bounds
2.5.2 Lower Bounds when Z is bounded
2.5.3 Proof of Theorem 2.1.6
2.A Appendix: Generalization to higher dimension
II Pinning models with constraints
3 An introduction to homogeneous and disordered Poland–Scheraga model
3.1 The Homogeneous Poland-Scheraga Model
3.2 The Disordered Poland-Scheraga Model
3.3 Beyond the Poland-Scheraga model
4 Homogeneous Circular DNA
Introduction
4.1 Circular DNA model(s)
4.1.1 A mathematical model
4.1.2 Notations and Main Assumptions
4.2 Main Results
4.2.1 Free Energy
4.2.2 Path Properties for the kernel models
4.2.3 Largest Jump for the constrained PS model
4.2.4 Discussion: phases and phase transitions in the kernel models
4.3 Proofs: free energy
4.3.1 Free Energy g(%)
4.3.2 Free Energy fH(h)
4.4 Proofs: largest jump for the constrained PS model
4.4.1 Proof in the “big jump domain”
4.4.2 Proof in the other regimes
4.5 Proofs: path properties for the models with a kernel
4.A Appendix: Results for heavy tail random walks
4.A.1 Slowly Varying Functions and Laplace Transform Estimates
4.A.2 Laplace Transform
4.A.3 Limit Theorems
4.A.4 Local Limit Theorems
5 Disordered Circular DNA
5.1 Models and Assumptions
5.2 Main Results
5.2.1 Free Energy: existence and regularity
5.2.2 Path Properties
5.3 Proofs: Free energies
5.3.1 Free energy g(; %)
5.3.2 Free energy f(; h) and fH(; h)
5.3.3 Free energy regularity
5.4 Proofs: path properties
5.5 On strict convexity for the PS free energy
Bibliography

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