A Course on Large Deviations with an
                    Introduction to Gibbs Measures

Rassoul-Agha, Firas; Seppäläinen, Timo

doi:10.1090/gsm/162

Cited by 123 publications

(124 citation statements)

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“…As it depends only on the neighborhood N , we denote the resulting Lyapunov profile by L N . By elementary large deviations (Dembo and Zeitouni 1998;Rassoul-Agha and Seppäläinen 2015), we can give it as a variational formula. For y ∈ R d , let…”

Section: Lyapunov Profilesmentioning

confidence: 99%

“…1 Evolution of defect percolation CA up to time 100 (red sites are those that contain at least one defect) and empirical Lyapunov profiles at time 10 5 for rules 7, 22, 38, and 110. These profiles encapsulate the exponential accumulation rate versus space-time direction (Color figure online) 1992) is a well-established subfield of the large deviations theory (Dembo and Zeitouni 1998;Rassoul-Agha and Seppäläinen 2015). The main idea is that the resulting profiles are given by a variational method: the process seeks the most advantageous option for accumulation at a spatial location; in general, the search space can have a very high dimension.…”

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confidence: 99%

“…Study of the stability of periodic solutions of dynamical systems also has a long history and is known as Floquet theory; see, e.g., Moore (2004) for a recent perspective. We are able to develop a fairly complete analog for CA dynamics, based on large deviations for finite Markov chains (Dembo and Zeitouni 1998;Rassoul-Agha and Seppäläinen 2015). These methods work particularly well under the irreducibility assumption, in which case the Lyapunov profile is given by a one-dimensional variational problem.…”

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confidence: 99%

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Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles

Baetens

Gravner

2016

J Nonlinear Sci

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We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.

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Section: Lyapunov Profilesmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles

Baetens

Gravner

2016

J Nonlinear Sci

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“…These few examples indicate that that the language of large deviations is even more useful in the presence of disorder in order to describe within a unified perspective all these phenomena involving typical and rare events on various scales.The aim of the present pedagogical introduction is thus to explain to physicists how the general theory of large deviations is the natural language to analyze the properties of various well-known classical and quantum random models. It is of course not meant for mathematicians who have been using the large deviation framework for a very long time (see the books [39][40][41][42][43][44] and references therein), in particular in the area of disordered systems (see the the books [45][46][47], the review [48] and references therein). This pedagogical introduction is thus intended only for physicists who are disheartened by the technical vocabulary used in the mathematical literature on large deviations (like Polish space, Borel sigma-field, cadlag function, ... ).The following sections are organized as follows.…”

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Revisiting classical and quantum disordered systems from the unifying perspective of large deviations

Monthus

2019

Eur. Phys. J. B

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The theory of large deviations is already the natural language for the statistical physics of equilibrium and non-equilibrium. In the field of disordered systems, the analysis via large deviations is even more useful to describe within a unified perspective the typical events and the rare events that occur on various scales. In the present pedagogical introduction, we revisit various emblematic classical and quantum disordered systems in order to highlight the common underlying mechanisms from the point of view of large deviations. I. INTRODUCTIONJust like Mr Jourdain discovering that he has been speaking in prose all his life without knowing it, physicists working in statistical physics become aware at some point that they have been using the theory of large deviations without realizing it since their very first acquaintance with the Boltzmann notion of entropy and the Gibbs theory of ensembles. This language of large deviations has turned out to be very powerful to unify the statistical physics of equilibrium, non-equilibrium and dynamical systems (see the reviews [1-3] and references therein) and to formulate an appropriate statistical physics approach of dynamical trajectories for various Markovian processes (see the reviews [4-10] and the PhD Theses [11][12][13][14] and the HDR Thesis [15]).In the field of disordered systems, the presence of random disorder variables induce a lot of subtle effects for the probabilities of interesting observables. Physicists have understood from the very beginning that some observables are non-self-averaging, i.e. their disorder-averaged value is completely different from their typical value (see the books [16,17] and references therein). It was also realized very early that in each large typical sample, there will nevertheless occur rare anomalous regions of a certain size that may dominate some observables : famous examples are the Lifshitz essential singularities of the density of states near spectrum edges in Anderson localization models [16,[18][19][20][21], the Griffiths singularities for the statics [22,23] and the dynamics [24-26] of random classical models, and the Griffiths phases in random quantum models (see the reviews [27,28] and references therein). Finally at critical points, it was found that multifractal properties appear, for instance for the inverse participation ratios of eigenfunctions at Anderson localization transitions (see the reviews [29, 30] and references therein) or for correlation functions in random classical spin models [31][32][33][34][35][36][37][38], while at Infinite Disorder fixed points, many observables are even more broadly distributed [27,28]. These few examples indicate that that the language of large deviations is even more useful in the presence of disorder in order to describe within a unified perspective all these phenomena involving typical and rare events on various scales.The aim of the present pedagogical introduction is thus to explain to physicists how the general theory of large deviations is the natural language to analyz...

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“…[RAS09], and [LS15] for an extension to the case of tagged point processes), and we recast its properties in the setting of signed point processes in Section 2.6. We may then define the specific relative entropy of P ∈ P inv (Λ × X ) as (1.8) ent(P ) :=ˆΛ ent[P x ]dx.…”

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Large Deviations for the Two-Dimensional Two-Component Plasma

Leblé

Serfaty

Zeitouni

2016

Commun. Math. Phys.

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Abstract. We derive a large deviations principle for the two-dimensional two-component plasma in a box. As a consequence, we obtain a variational representation for the free energy, and also show that the macroscopic empirical measure of either positive or negative charges converges to the uniform measure. An appendix, written by Wei Wu, discusses applications to the supercritical complex Gaussian multiplicative chaos.

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A Course on Large Deviations with an Introduction to Gibbs Measures

Cited by 123 publications

References 18 publications

Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles

Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles

Revisiting classical and quantum disordered systems from the unifying perspective of large deviations

Large Deviations for the Two-Dimensional Two-Component Plasma

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