Sums of correlated exponentials: two types of Gaussian correlation structures

A. Klymovskiy

    Research output: ThesisPhd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

    Abstract

    In this thesis, we study the limiting behaviour of the large sums of strongly correlated exponentials as the number of their summands and the effective dimension of the correlation structure simultaneously tend to infinity.We consider two types of such sums which are generated by two a priori very different Gaussian correlation structures. The first type is a sum of hierarchically correlated random variables which is based on the partition function of Derrida’s generalised random energy model (GREM) with external field. The second type is an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick (SK) model with multidimensional spins. We consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations (i.e., at the level of weak limiting laws). Interestingly for the SK model with multidimensional spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports the conjectured in theoretical physics universal behaviour of the sums of such sort. Concerning the SK model with multidimensional spins, we obtain the following results. We prove upper and lower bounds on the free energy of this model in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional.We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle’s probability cascades. For this purpose, an abstract quenched large deviations principle of the G¨artner-Ellis type is obtained. We derive Talagrand’s representation of Guerra’s remainder term for the SK model with multidimensional spins. The derivation is based on well-known properties of Ruelle’s probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which, however, do not cover the original setup of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand’s methodology of a priori estimates. Concerning the GREM in the presence of uniform external field, we obtain the following results. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetisation, as prescribed by strength of external field and by parameters of the GREM.
    Original languageEnglish
    QualificationDoctor of Philosophy
    Awarding Institution
    • Technische Universität Berlin
    Supervisors/Advisors
    • Bovier, A., Promotor, External person
    • Bolthausen, E., Promotor, External person
    Award date24 Jun 2008
    Place of PublicationBerlin
    Publisher
    Publication statusPublished - 2008

    Fingerprint

    Correlation Structure
    Energy Model
    Free Energy
    External Field
    Thermodynamic Limit
    Fluctuations
    Partition Function
    Cascade
    Model
    Tend
    Strict Convexity
    Effective Dimension
    Sums of Random Variables
    Semilinear Differential Equations
    Coarse-graining
    Large Deviation Principle
    Limiting Behavior
    Linear partial differential equation
    Hierarchical Structure
    Error term

    Cite this

    Klymovskiy, A.. / Sums of correlated exponentials: two types of Gaussian correlation structures. Berlin : TU Berlin, 2008. 169 p.
    @phdthesis{bbe1e4e843f9431383eb897ad038ddcd,
    title = "Sums of correlated exponentials: two types of Gaussian correlation structures",
    abstract = "In this thesis, we study the limiting behaviour of the large sums of strongly correlated exponentials as the number of their summands and the effective dimension of the correlation structure simultaneously tend to infinity.We consider two types of such sums which are generated by two a priori very different Gaussian correlation structures. The first type is a sum of hierarchically correlated random variables which is based on the partition function of Derrida’s generalised random energy model (GREM) with external field. The second type is an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick (SK) model with multidimensional spins. We consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations (i.e., at the level of weak limiting laws). Interestingly for the SK model with multidimensional spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports the conjectured in theoretical physics universal behaviour of the sums of such sort. Concerning the SK model with multidimensional spins, we obtain the following results. We prove upper and lower bounds on the free energy of this model in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional.We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle’s probability cascades. For this purpose, an abstract quenched large deviations principle of the G¨artner-Ellis type is obtained. We derive Talagrand’s representation of Guerra’s remainder term for the SK model with multidimensional spins. The derivation is based on well-known properties of Ruelle’s probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which, however, do not cover the original setup of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand’s methodology of a priori estimates. Concerning the GREM in the presence of uniform external field, we obtain the following results. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetisation, as prescribed by strength of external field and by parameters of the GREM.",
    author = "A. Klymovskiy",
    year = "2008",
    language = "English",
    publisher = "TU Berlin",
    school = "Technische Universit{\"a}t Berlin",

    }

    Klymovskiy, A 2008, 'Sums of correlated exponentials: two types of Gaussian correlation structures', Doctor of Philosophy, Technische Universität Berlin, Berlin.

    Sums of correlated exponentials: two types of Gaussian correlation structures. / Klymovskiy, A.

    Berlin : TU Berlin, 2008. 169 p.

    Research output: ThesisPhd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

    TY - THES

    T1 - Sums of correlated exponentials: two types of Gaussian correlation structures

    AU - Klymovskiy, A.

    PY - 2008

    Y1 - 2008

    N2 - In this thesis, we study the limiting behaviour of the large sums of strongly correlated exponentials as the number of their summands and the effective dimension of the correlation structure simultaneously tend to infinity.We consider two types of such sums which are generated by two a priori very different Gaussian correlation structures. The first type is a sum of hierarchically correlated random variables which is based on the partition function of Derrida’s generalised random energy model (GREM) with external field. The second type is an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick (SK) model with multidimensional spins. We consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations (i.e., at the level of weak limiting laws). Interestingly for the SK model with multidimensional spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports the conjectured in theoretical physics universal behaviour of the sums of such sort. Concerning the SK model with multidimensional spins, we obtain the following results. We prove upper and lower bounds on the free energy of this model in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional.We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle’s probability cascades. For this purpose, an abstract quenched large deviations principle of the G¨artner-Ellis type is obtained. We derive Talagrand’s representation of Guerra’s remainder term for the SK model with multidimensional spins. The derivation is based on well-known properties of Ruelle’s probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which, however, do not cover the original setup of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand’s methodology of a priori estimates. Concerning the GREM in the presence of uniform external field, we obtain the following results. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetisation, as prescribed by strength of external field and by parameters of the GREM.

    AB - In this thesis, we study the limiting behaviour of the large sums of strongly correlated exponentials as the number of their summands and the effective dimension of the correlation structure simultaneously tend to infinity.We consider two types of such sums which are generated by two a priori very different Gaussian correlation structures. The first type is a sum of hierarchically correlated random variables which is based on the partition function of Derrida’s generalised random energy model (GREM) with external field. The second type is an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick (SK) model with multidimensional spins. We consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations (i.e., at the level of weak limiting laws). Interestingly for the SK model with multidimensional spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports the conjectured in theoretical physics universal behaviour of the sums of such sort. Concerning the SK model with multidimensional spins, we obtain the following results. We prove upper and lower bounds on the free energy of this model in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional.We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle’s probability cascades. For this purpose, an abstract quenched large deviations principle of the G¨artner-Ellis type is obtained. We derive Talagrand’s representation of Guerra’s remainder term for the SK model with multidimensional spins. The derivation is based on well-known properties of Ruelle’s probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which, however, do not cover the original setup of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand’s methodology of a priori estimates. Concerning the GREM in the presence of uniform external field, we obtain the following results. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetisation, as prescribed by strength of external field and by parameters of the GREM.

    M3 - Phd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

    PB - TU Berlin

    CY - Berlin

    ER -