Limit Of Infinite Dimensions Research Articles

Proof of a Conjecture on the Infinite Dimension Limit of a Unifying Model for Random Matrix Theory

We study the large $N$ limit of a sparse random block matrix ensemble. It depends on two parameters: the average connectivity $Z$ and the size of the blocks $d$, which is the dimension of an euclidean space. In the limit of large $d$, with $\frac{Z}{d}$ fixed, we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the Adjacency block matrix to the one of the effective medium approximation, in the case of the Laplacian block matrix to the Marchenko-Pastur distribution. We extend previous analytical computations of the moments of the spectral density of the Adjacency block matrix and the Lagrangian block matrix, valid for all values of $Z$ and $d$.

Dynamics around the site percolation threshold on high-dimensional hypercubic lattices.

Recent advances on the glass problem motivate reexamining classical models of percolation. Here we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, d_{u}=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥d_{u}. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit d→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d_{u}. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.

A Scaling Limit for Limit Order Books Driven by Hawkes Processes

In this paper we derive a scaling limit for an infinite-dimensional limit order book model driven by Hawkes random measures. The dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator. With our choice of scaling the dynamics converges to a coupled SDE-ODE system where limiting best bid and ask price processes follows a diffusion dynamics, the limiting volume density functions follows an ODE in a Hilbert space, and the limiting order arrival and cancellation intensities follow a Volterra--Fredholm integral equation.

Second order approximations for limit order books

In this paper, we derive a second order approximation for an infinite-dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g. the volume standing at the top of the book). We study the fluctuations of the price and volume process relative to their first order approximation given in ODE–PDE form under two different scaling regimes. In the first case, we suppose that price changes are really rare, yielding a constant first order approximation for the price. This leads to a measure-valued SDE driven by an infinite-dimensional Brownian motion in the second order approximation of the volume process. In the second case, we use a slower rescaling rate, which leads to a non-degenerate first order approximation and gives a PDE with random coefficients in the second order approximation for the volume process. Our results can be used to derive confidence intervals for models of optimal portfolio liquidation under market impact.

Emergence of Floquet behavior for lattice fermions driven by light pulses

As many-body Floquet theory becomes more popular, it is important to find ways to connect theory with experiment. Theoretical calculations can have a periodic driving field that is always on, but experiment cannot. Hence, we need to know how long a driving field is needed before the system starts to look like the periodically driven Floquet system. We answer this question here for noninteracting band electrons in the infinite-dimensional limit by studying the properties of the system under pulsed driving fields and illustrating how they approach the Floquet limit. Our focus is on determining the minimal pulse lengths needed to recover the qualitative and semiquantitative Floquet theory results.

Liu-Nagel phase diagrams in infinite dimension

We study Harmonic Soft Spheres as a model of thermal structural glasses in the limit of infinite dimension. We show that cooling, compressing and shearing a glass lead to a Gardner transition and, hence, to a marginally stable amorphous solid as found for Hard Spheres systems. A general outcome of our results is that a reduced stability of the glass favors the appearance of the Gardner transition. Therefore using strong perturbations, e.g. shear and compression, on standard glasses or using weak perturbations on weakly stable glasses, e.g. the ones prepared close to the jamming point, are the generic ways to induce a Gardner transition. The formalism that we discuss allows to study general perturbations, including strain deformations that are important to study soft glassy rheology at the mean field level.

The binomial decomposition of OWA functions, the 2-additive and 3-additive cases inndimensions

In the context of the binomial decomposition of ordered weighted averaging (OWA) functions, we investigate the constraints associated with the 2-additive and 3-additive cases in n dimensions. The 2-additive case depends on one coefficient whose feasible region does not depend on the dimension n. On the other hand, the feasible region of the 3-additive case depends on two coefficients and is explicitly dependent on the dimension n. This feasible region is a convex polygon with n vertices and n edges, which is strictly expanding in the dimension n. The orness of the OWA functions within the feasible region is linear in the two coefficients, and the vertices associated with maximum and minimum orness are identified. Finally, we discuss the 3-additive binomial decomposition in the asymptotic infinite dimensional limit.

Strange metal from Gutzwiller correlations in infinite dimensions: Transverse transport, optical response, and rise of two relaxation rates

Using two approaches to strongly correlated systems, the extremely correlated Fermi liquid theory and the dynamical mean field theory, we compute the transverse transport coefficients, namely the Hall constants $R_H$ and Hall angles $\theta_H$, and also the longitudinal and transverse optical response of the $U=\infty$ Hubbard model in the limit of infinite dimensions. We focus on two successive low-temperature regimes, the Gutzwiller correlated Fermi liquid (GCFL) and the Gutzwiller correlated strange metal (GCSM). We find that the Hall angles $\cot \theta_H \propto T^2$ in the GCFL regime, on warming into the strange metal regime, it passes through a downward bend and continues as $T^2$. Equivalently, $R_H$ is weakly temperature dependent in the GCFL regime, and becomes strongly $T$-dependent in the GCSM regime. Drude peaks are found for both the longitudinal optical conductivity $\sigma_{xx}(\omega)$ and the optical Hall angles $\tan \theta_H(\omega)$ below certain characteristic energy scales. By comparing the relaxation rates extracted from fitting to the Drude formula, we find that in the GCFL regime there is a single relaxation rate controlling both longitudinal and transverse transport, while in the GCSM regime two independent relaxation rates emerge. We trace the origin of this behavior to the dynamical particle-hole asymmetry of the Dyson self-energy, arguably a generic feature of doped Mott insulators.

Formula omitted]QIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit

formula omitted]QIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit

Strong-disorder approach for the Anderson localization transition

We implement an efficient strong-disorder renormalization-group (SDRG) procedure to study disordered tight-binding models in any dimension and on the Erdos-Renyi random graphs, which represent an appropriate infinite dimensional limit. Our SDRG algorithm is based on a judicious elimination of most (irrelevant) new bonds generated under RG. It yields excellent agreement with exact numerical results for universal properties at the critical point without significant increase of computer time, and confirm that, for Anderson localization, the upper critical dimension duc = infinite. We find excellent convergence of the relevant 1/d expansion down to d=2, in contrast to the conventional 2+epsilon expansion, which has little to say about what happens in any d>3. We show that the mysterious mirror symmetry of the conductance scaling function is a genuine strong-coupling effect, as speculated in early work. This opens an efficient avenue to explore the critical properties of Anderson transition in the strong-coupling limit in high dimensions.

Real Space Migdal–Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment

In this manuscript, in honour of L. Kadanoff, we present recent progress obtained in the description of finite dimensional glassy systems thanks to the Migdal-Kadanoff renormalisation group (MK-RG). We provide a critical assessment of the method, in particular discuss its limitation in describing situations in which an infinite number of pure states might be present, and analyse the MK-RG flow in the limit of infinite dimensions. MK-RG predicts that the spin-glass transition in a field and the glass transition are governed by zero-temperature fixed points of the renormalization group flow. This implies a typical energy scale that grows, approaching the transition, as a power of the correlation length, thus leading to enormously large time-scales as expected from experiments and simulations. These fixed points exist only in dimensions larger than $d_L>3$ but they nevertheless influence the RG flow below it, in particular in three dimensions. MK-RG thus predicts a similar behavior for spin-glasses in a field and models of glasses and relates it to the presence of avoided critical points.

Role of Ito's lemma in sampling pinned diffusion paths in the continuous-time limit.

We consider pinned diffusion paths that are explored by a particle moving via a conservative force while being in thermal equilibrium with its surroundings. To probe rare transitions, we use the Onsager-Machlup (OM) functional as a path probability distribution function for transition paths that are constrained to start and stop at predesignated points in different energy basins after a fixed time. The OM theory is based on a discrete-time version of Brownian dynamics, and thus it possesses a finite number of time steps. Here we explore the continuous-time limit where the number of time steps, and hence the dimensionality, becomes infinite. In this regime, the OM functional has been commonly regularized by using the Ito-Girsanov change of measure. This regularized form can then be used as a basis of a numerical algorithm to probe transition paths. In doing so, time again is discretized, progressing in fixed increments. When sampling paths, we find that numerical schemes based on this regularized continuous-time limit can fail catastrophically in describing the path of a particle moving in a potential with multiple wells. The origin of this behavior is traced to numerical instabilities in the discrete version of the continuous-time path measure that are not present in the infinite-dimensional limit. These instabilities arise because of the difficulty of satisfying, in finite dimensions, the conditions imposed by Ito's lemma that was an essential ingredient in the derivation of the regularized continuous-time measure. As an important consequence of this analysis, we conclude that the most probable diffusion path is not a physical entity because the thermodynamic action is effectively flat and cannot be minimized.

Reminiscences, and some explorations about the bootstrap

The paper is a potpourri of short sections. There will be some reminiscences about Evarist (from the early 1970s), then some on infinite-dimensional limit theorems from 1950 through 1990. A section reviews a case of slow convergence in the central limit theorem for empirical processes (Beck, 1985) and another the “fast” convergence of Komlós–Major–Tusnády. The paper does an experimental exploration of bootstrap confidence intervals for the mean (of Pareto distributions) and (as less commonly seen) for the variance, of normal and Pareto distributions.

Assessing the orbital selective Mott transition with variational wave functions

We study the Mott metal–insulator transition in the two-band Hubbard model with different hopping amplitudes t1 and t2 for the two orbitals on the two-dimensional square lattice by using non-magnetic variational wave functions, similarly to what has been considered in the limit of infinite dimensions by dynamical mean-field theory. We work out the phase diagram at half filling (i.e. two electrons per site) as a function of and the on-site Coulomb repulsion U, for two values of the Hund’s coupling J = 0 and J/U = 0.1. Our results are in good agreement with previous dynamical mean-field theory calculations, demonstrating that the non-magnetic phase diagram is only slightly modified from infinite to two spatial dimensions. Three phases are present: a metallic one, for small values of U, where both orbitals are itinerant; a Mott insulator, for large values of U, where both orbitals are localized because of the Coulomb repulsion; and the so-called orbital-selective Mott insulator (OSMI), for small values of R and intermediate Us, where one orbital is localized while the other one is still itinerant. The effect of the Hund’s coupling is two-fold: on one side, it favors the full Mott phase over the OSMI; on the other side, it stabilizes the OSMI at larger values of R.

Born-Infeld gravity with a unique vacuum and a massless graviton

We construct an n-dimensional Born-Infeld type gravity theory that has the same properties as Einstein's gravity in terms of the vacuum and particle content: Namely, the theory has a unique viable vacuum (maximally symmetric solution) and a single massless unitary spin-2 graviton about this vacuum. The BI gravity, in some sense, is the most natural, minimal generalization of Einstein's gravity with a better UV behavior, and hence, is a potentially viable proposal for low energy quantum gravity. The Gauss-Bonnet combination plays a non-trivial role in the construction of the theory. As an extreme example, we consider the infinite dimensional limit where an interesting exponential gravity arises.

One-dimensional Kac model of dense amorphous hard spheres

We introduce a new model of hard spheres under confinement for the study of the glass and jamming transitions. The model is a one-dimensional chain of the d-dimensional boxes each of which contains the same number of hard spheres, and the particles in the boxes of the ends of the chain are quenched at their equilibrium positions. We focus on the infinite-dimensional limit of the model and analytically compute the glass transition densities using the replica liquid theory. From the chain length dependence of the transition densities, we extract the characteristic length scales at the glass transition. The divergence of the lengths are characterized by the two exponents, for the dynamical transition and −1 for the ideal glass transition, which are consistent with those of the p-spin mean-field spin glass model. We also show that the model is useful for the study of the growing length scale at the jamming transition.

Uncertainty Relations for General Canonically Conjugate Observables in Terms of Unified Entropies

We study uncertainty relations for a general class of canonically conjugate observables. It is known that such variables can be approached within a limiting procedure of the Pegg–Barnett type. We show that uncertainty relations for conjugate observables in terms of generalized entropies can be obtained on the base of genuine finite-dimensional consideration. Due to the Riesz theorem, there exists an inequality between norm-like functionals of two probability distributions in finite dimensions. Using a limiting procedure of the Pegg–Barnett type, we take the infinite-dimensional limit right for this inequality. Hence, uncertainty relations are derived in terms of generalized entropies. In particular, the case of measurements with a finite precision is addressed. It takes into account that in any experiment an accuracy of performed measurements is always limited.

Finiteness results for Abelian tree models

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant § refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial and the tree is a star, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models.

Gardner transition in finite dimensions

Recent works on hard spheres in the limit of infinite dimensions revealed that glass states, envisioned as metabasins in configuration space, can break up in a multitude of separate basins at low enough temperature or high enough pressure, leading to the emergence of new kinds of soft-modes and unusual properties. In this paper we study by perturbative renormalization group techniques the critical properties of this transition, which has been discovered in disordered mean-field models in the 1980s. We find that the upper-critical dimension ${d}_{u}$, above which mean-field results hold, is strictly larger than six and apparently nonuniversal, i.e., system dependent. Below ${d}_{u}$, we do not find any perturbative attractive fixed point (except for a tiny region of the one-step replica symmetry breaking parameter), thus showing that the transition in three dimensions either is governed by a nonperturbative fixed point unrelated to the Gaussian mean-field one or becomes first order or does not exist. We also discuss possible relationships with the behavior of spin glasses in a field.

Bohr’s molecular model and the melding of classical and quantum mechanics

Svidzinsky, Scully, and Herschbach reply: We agree with M. Y. Amusia that a statement we made about the infinite dimension limit is “lacking in rigor.” It was intended simply to provide heuristic insight into why, for a wide variety of problems, the large-D limit has proven to be a useful starting approximation to obtain results for D = 3.Amusia’s other concerns are answered in a 1975 paper on a D-scaling treatment of helium by David Herrick and Frank Stillinger.11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 They gave a rigorous derivation of the D-dimensional hydrogen atom Hamiltonian, shown as equation 1 in Amusia’s letter. That solution applies for D ≥ 2, as implied in the figure included in box 1 of our article. Herrick and Stillinger also showed that for the correct D → 1 limit, the Z/r term becomes a δ-function. In D-scaling, contrary to Amusia’s assumption, the D = 3 form of Coulomb’s law can be used for dimensional continuation to the large-D limit. The D-dependent similarity transformation affects the Laplacian, not the potential energy. Both theory and application are amply presented in references given in our article (particularly references 8–11). D-scaling, using just elementary algebra, has attained correlation energies for multielectron atoms with accuracy comparable to or better than conventional electronic calculations.22. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059D. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5The papers of Leonard Mlodinow33. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584 and several other authors, especially the earlier paper by Herrick and Stillinger,11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 fostered the development of D-scaling for electronic structure. Since the kinship of Bohr’s model to dimensional scaling was not recognized until 2005, we did not dwell on that history, other than citing the tutorial article by Edward Witten. The treatment of H2 that Mlodinow cites in the second part of his reference 3 is a deliberately drastic approximation and gives less than 40% of the bond dissociation energy.As emphasized by Petar Grujic, the bold, perplexing enterprise by Niels Bohr motivated much further work melding classical and quantum mechanics. We note that D-scaling has a distinctive character. It might aptly be termed “semiquantum” rather than semiclassical. Although in the large-D limit the equations become classical, quantum mechanics is hidden in the D-dependent units adopted for distance and energy. At that limit, electrons take fixed positions in the D-scaled space and the first-order correction in (1/D) has them execute harmonic vibrations about those positions. In the prequantum era, such behavior was postulated by Gilbert Lewis and Irving Langmuir,44. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. motivated by chemical arguments but disdained by physicists and considered incompatible with the Bohr model.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42, Google ScholarCrossref, ISI2. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059, Google ScholarCrossref, ISID. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5, , Google ScholarCrossref, ISI3. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584, Google ScholarCrossref, ISI4. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. Google Scholar© 2014 American Institute of Physics.