Littérature scientifique sur le sujet « Random polymers, Universality, Weak disorder »

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

Isotropic-genesis nematic elastomers (IGNEs) are liquid crystalline polymers (LCPs) that have been randomly, permanently cross-linked in the high-temperature state so as to form an equilibrium random solid. Thus, instead of being free to diffuse throughout the entire volume, as they would be in the liquid state, the constituent LCPs in an IGNE are mobile only over a finite, segment specific, length-scale controlled by the density of cross-links. We address the effects that such network-induced localization have on the liquid–crystalline characteristics of an IGNE, as probed via measurements made at high temperatures. In contrast with the case of uncross-linked LCPs, for IGNEs these characteristics are determined not only by thermal fluctuations but also by the quenched disorder associated with the cross-link constraints. To study IGNEs, we consider a microscopic model of dimer nematogens in which the dimers interact via orientation-dependent excluded volume forces. The dimers are, furthermore, randomly, permanently cross-linked via short Hookean springs, the statistics of which we model by means of a Deam–Edwards type of distribution. We show that at length-scales larger than the size of the nematogens this approach leads to a recently proposed, phenomenological Landau theory of IGNEs [Lu et al., Phys. Rev. Lett.108, 257803 (2012)], and hence predicts a regime of short-ranged oscillatory spatial correlations in the nematic alignment, of both thermal and glassy types. In addition, we consider two alternative microscopic models of IGNEs: (i) a wormlike chain model of IGNEs that are formed via the cross-linking of side-chain LCPs; and (ii) a jointed chain model of IGNEs that are formed via the cross-linking of main-chain LCPs. At large length-scales, both of these models give rise to liquid–crystalline characteristics that are qualitatively in line with those predicted on the basis of the dimer-and-springs model, reflecting the fact that the three models inhabit a common universality class.

Da un punto di vista chimico e fisico, un polimero è una lunga catena di unità ripetute, chiamate monomeri, quasi identiche nella struttura, ma che possono differire tra loro per il grado di affinità rispetto ad alcuni solventi. Questa caratteristica permette di avere delle interazioni tra il polimero e l’ambiente in cui esso si trova. Tale interazione può dare luogo a fenomeni di localizzazione e concentrazione ed è possibile osservare una transizione di fase. Nel caso in cui questa regione è un punto o una linea si parla di modello di pinning, che rappresenta il principale oggetto di studio di questa tesi. Matematicamente il modello di pinning descrive il comportamento di una catena di Markov in interazione con uno suo stato dato. Questa interazione può attirare o respingere il cammino della catena di Markov con una forza modulata da due parametri, h e β. Quando β = 0 si parla di modello omogeneo, che è completamente risolubile. Il modello disordinato, i.e., β > 0, è matematicamente più interessante. In questo caso l'interazione dipende da un parametro aleatorio esterno, indipendente dalla catena di Markov, chiamato disordine. L’interazione è realizzata modificando la legge originale della catena di Markov attraverso una misura di Gibbs (dipendente dal disordine, h e β). L’obiettivo principale è comprendere la struttura della catena di Markov rispetto a questa nuova probabilità. Il primo lavoro di ricerca di questa tesi riguarda il modello di pinning in cui si considera un disordine a code pesanti e il tempo di ritorno della catena di Markov avente una distribuzione sub-esponenziale. Noi dimostriamo che l’insieme dei tempi in cui la catena di Markov visita lo stato dato, opportunamente riscalato, converge in legge verso un insieme limite, dipendente unicamente dal disordine. Dimostriamo inoltre che esiste una transizione di fase con un punto critico aleatorio sotto il quale l’insieme limite è banale. In un secondo lavoro consideriamo il modello di pinning con un disordine a code leggere e il tempo di ritorno della catena di Markov con una distribuzione a code polinomiali di esponente 0 < α < 1. Sotto queste ipotesi si può dimostrare che esiste un’interazione non banale tra i parametri h e β che dà origine a un punto critico, h_c (β), dipendente unicamente dalle leggi del disordine e della catena di Markov. Se h > h_c (β), allora la catena di Markov è localizzata attorno allo stato dato e lo visita un numero infinito di volte. Altrimenti, se h < h_c (β), la catena di Markov è delocalizzata. Un problema molto interessante riguarda il comportamento del modello nel limite del disordine debole: più precisamente vogliamo comprendere il comportamento del punto critico h_c(β) quando β → 0. Nel caso 1/2 < α < 1, in letteratura non esistono stime precise sull’asintotica di h_c (β) per β → 0. Aver trovato l’asintotica precisa rappresenta il risultato più importante di questa tesi.
A polymer is a long chain of repeated units (monomers) that are almost identical, but they can differ in their degree of affinity for certain solvents. Such property allows to have interactions between the polymer and the external environment. This interaction can attract or repel the polymer, giving rise to localization and concentration phenomena. It is then possible to observe the existence of a phase transition. Whenever such region is a point or a line, then we talk about pinning model, which represents the main subject of this thesis. From a mathematical point of view, the pinning model describes the behavior of a Markov chain in interaction with a distinguished state. This interaction can attract or repel the Markov chain path with a force tuned by two parameters, h and β. If β = 0 we obtain the homogeneous pinning model, which is completely solvable. The disordered pinning model, which corresponds to β > 0, is most challenging and mathematically interesting. In this case the interaction depends on an external source of randomness, independent of the Markov chain, called disorder. The interaction is realized by perturbing the original Markov chain law via a Gibbs measure, which depends on the disorder, hand β. Our main aim is to understand the structure of the Markov chain under this new probability measure. The first research topic of this thesis is the pinning model in which the disorder is heavy-tailed and the return times of the Markov chain have a sub-exponential distribution. We prove that the set of the times at which the Markov chain visits the distinguished state, suitably rescaled, converges in distribution to a limit random set which depends only on the disorder. We show that there exists a phase transition with a random critical point, below which the limit set is trivial. In the second part of the thesis We consider a pinning model with a light-tailed disorder and the return times of the Markov chain with a polynomial tail distribution, with exponent 0 < α < 1. It is possible to show that there exists a non-trivial interaction between the parameters h and β. Such interaction gives rise to a critical point, hc (β), depending only on the law of the disorder and of the Markov chain. If h > hc (β), then the Markov chain visits infinitely many times the distinguished state and we say that it is localized. Otherwise, if h < hc (β), then the Markov chain visits such state only a finite number of times. Therefore the critical behavior of the model is deeply connected with the structure of hc (β). A very challenging problem is to describe the behavior of the pinning model in the weak disorder regime. To be more precise, one wants to understand the behavior of the critical point when β → 0. In the case of 1/2 < α < 1, in the literature there are non-matching estimates about the asymptotics of hc (β) as β → 0. Getting the exact asymptotics for hc (β) represents the most important result of this thesis.

Le modèle d'accrochage de polymère décrit le comportement d'une chaîne de Markov en interaction avec un état donné. Cette interaction peut attirer ou repousser la chaîne de Markov et elle est modulée par deux paramètres, h et β. Quand β = 0 on parle de modèle homogène, qui est complètement solvable. Le modèle désordonné, i.e. quand β > 0, est mathématiquement le plus intéressant. Dans ce cas, l'interaction dépend d'une source d'aléa extérieur indépendant de la chaîne de Markov, appelée désordre. L'interaction est réalisée en modifiant la loi originelle de la chaîne de Markov par une mesure de Gibbs et la probabilité obtenue définit le modèle d'accrochage de polymère. Le but principal est d'étudier et de comprendre la structure des trajectoires typiques de la chaîne de Markov sous cette nouvelle probabilité. Le premier sujet de recherche concerne le modèle d'accrochage de polymère où le désordre est à queues lourdes et où le temps de retour de la chaîne de Markov suit une distribution sous-exponentielle. Dans notre deuxième résultat nous étudions le modèle d'accrochage de polymère avec un désordre à queues légères et le temps de retour de la chaîne de Markov avec une distribution à queues polynomiales d'exposant α > 0. On peut démontrer qu'il existe un point critique, h(β). Notre but est comprendre le comportement du point critique quand β -> 0. La réponse dépend de la valeur de α. Dans la littérature on a des résultats précis pour α < ½ et α > 1. Nous montrons que α ∈ (1/2, 1) le comportement du modèle dans la limite du désordre faible est universel et le point critique, opportunément changé d'échelle, converge vers la même quantité donnée par un modèle continu
The pinning model describes the behavior of a Markov chain in interaction with a distinguished state. This interaction can attract or repel the Markov chain path with a force tuned by two parameters, h and β. If β = 0 we obtain the homogeneous pinning model, which is completely solvable. The disordered pinning model, i.e. when β > 0, is most challenging and mathematically interesting. In this case the interaction depends on an external source of randomness, independent of the Markov chain, called disorder. The interaction is realized by perturbing the original Markov chain law via a Gibbs measure, which defines the Pinning Model. Our main aim is to understand the structure of a typical Markov chain path under this new probability measure. The first research topic of this thesis is the pinning model in which the disorder is heavy-tailed and the return times of the Markov chain have a sub-exponential distribution. In our second result we consider a pinning model with a light-tailed disorder and the return times of the Markov chain with a polynomial tail distribution, with exponent α > 0. It is possible to show that there exists a critical point, h(β). Our goal is to understand the behavior of the critical point when β -> 0. The answer depends on the value of α and in the literature there are precise results only for the case α < ½ et α > 1. We show that for α ∈ (1/2, 1) the behavior of the pinning model in the weak disorder limit is universal and the critical point, suitably rescaled, converges to the related quantity of a continuum model

This chapter covers the one-dimensional Kardar–Parisi–Zhang equation, weak drive limit, universality, directed polymers in a random medium, replica solutions, statistical mechanics of line ensembles, and its generalization to several components which is used to study equilibrium time correlations of anharmonic chains and of the discrete nonlinear Schrödinger equation.