Infinite Limit Research Articles

Ising Model on Random Triangulations of the Disk: Phase Transition

In Chen and Turunen (Commun Math Phys 374(3):1577–1643, 2020), we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in Chen and Turunen (2020) to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity).

Geometric structure of thermal cones.

The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future, and incomparable regions. In this work, we analyze the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a d-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behavior based on thermodynamic monotones given by the volumes of thermal cones. Results obtained apply also to other majorization-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit. Finally, we also generalize the construction of thermal cones to account for probabilistic transformations.

Metaphorical Transfers in the Azerbaijani and English Languages in Publicistic Texts

<p><strong><em>The purpose of the study</em></strong><em> is to identify current trends in metaphorical transfers in publicistic texts in Azerbaijani and English languages.</em><em></em></p><p><strong><em>Research methodology.</em></strong><em> In the course of the research, during the elaboration of the theoretical achievements of scholars the methods of analysis, synthesis, generalization, systematization were used; the method of comparison was used for identification of common and different in journalistic texts in Azerbaijani and English. The study is based on the theory of cognitive linguistics, according to which transfers are both linguistic and philosophical acts.</em></p><p><strong><em>Results.</em></strong><em> The article deals with metaphorical transfers in Azerbaijani and English. They apply to all people, regardless of their race or nationality. Linguistic sources of the outlined subject are analyzed. Human consciousness is able to maintain an infinite system of associations. As a rule, the conceptualization of different branches of knowledge based on metaphorical transfer is based on logical-syntactic and semantic frameworks of language. Metaphorical transfers, according to the theory of cognitive linguistics, are both linguistic and philosophical actions. The rationale for such provisions is presented in Western linguistics, modern research has confirmed that metaphorization is the result of intellectual activity, and its manifestation is the structure of the language. As for the human imagination, which usually has infinite limits, it is able to create the most complex cognitive models. These models of transfer, which are part of cultural thinking, are expressed through language. At the same time, people share the same cognitive models because they have similar life experiences. The article shows that metaphorical transfers in Azerbaijani and English languages are practically indistinguishable or have slight variations.</em></p><p><strong><em>Novelty.</em></strong><em> It is demonstrated by concrete examples that in the minds of all people there is a common metaphorical model and that forms of expression change depending on the interlocutors.</em></p><p><strong><em>Practical importance.</em></strong><em> The results obtained may be of interest to media language researchers and lecturers of philology and journalism.</em></p><p><strong><em>Key</em></strong><strong><em> </em></strong><strong><em>words</em></strong><em>: English language, Azerbaijani language, cognitive linguistics, metaphorical transfer, publicistic text.</em></p>

Profiles of future mathematics teachers in the teaching of the infinite limit of sequences

Resumen En este estudio se pretenden describir perfiles del futuro docente de matemáticas en relación con la enseñanza del límite infinito de una sucesión. Para ello, se conformaron grupos focales en los que el futuro profesorado dialogó sobre la manera en la que se abordaría la enseñanza de este límite y, posteriormente, se estableció un análisis de coincidencias reticular, con el que presentar la aparición de un conjunto de sucesos mediante un grafo. Los resultados arrojados permitieron determinan cómo el profesorado en formación empleará el enfoque intuitivo y formal en sus comentarios sobre la enseñanza del límite infinito de una sucesión. Además, analizamos si la formación es una variable a tener en cuenta en las respuestas de los futuros docentes a tareas relacionadas con el límite infinito.

Integrodifferential equation in the multimode Jaynes-Cummings model

We investigate the multimode Jaynes-Cummings model. The population of the excited state of an atom is expressed by an integrodifferential equation. The exact solution is found in an idealized case by taking the infinite limit of the number of modes. The solution shows a mathematical reason why the kinklike behavior of the excited-state population appears. We also confirm that numerical calculations agree with the exact solution.

The infinite Schmidt number limit of the salt fingering convection model and the inertial free salt convection model

We show that the large Schmidt number limit with fixed Prandtl number and Rayleigh number ratio of the salt convection model is exactly the inertial free salt convection model (IFSC). The limit is a singular one, with simultaneous perturbation in both the velocity and temperature field, and the rigorous mathematical justification involves two-time scale expansion.

Do ideas have shape? Idea registration as the continuous limit of artificial neural networks

We introduce a Gaussian Process (GP) generalization of ResNets (with unknown functions of the network replaced by GPs and identified via MAP estimation), which includes ResNets (trained with L2 regularization on weights and biases) as a particular case (when employing particular kernels). We show that ResNets (and their warping GP regression extension) converge, in the infinite depth limit, to a generalization of image registration variational algorithms. In this generalization, images are replaced by functions mapping input/output spaces to a space of unexpressed abstractions (ideas), and material points are replaced by data points. Whereas computational anatomy aligns images via warping of the material space, this generalization aligns ideas (or abstract shapes as in Plato’s theory of forms) via the warping of the Reproducing Kernel Hilbert Space (RKHS) of functions mapping the input space to the output space. While the Hamiltonian interpretation of ResNets is not new, it was based on an Ansatz. We do not rely on this Ansatz and present the first rigorous proof of convergence of ResNets with trained weights and biases towards a Hamiltonian dynamics driven flow. Since our proof is constructive and based on discrete and continuous mechanics, it reveals several remarkable properties of ResNets and their GP generalization. ResNets regressors are kernel regressors with data-dependent warping kernels. Minimizers of L2 regularized ResNets satisfy a discrete least action principle implying the near preservation of the norm of weights and biases across layers. The trained weights of ResNets with scaled/strong L2 regularization can be identified by solving an autonomous Hamiltonian system. The trained ResNet parameters are unique up to (a function of) the initial momentum, and the initial momentum representation of those parameters is generally sparse. The kernel (nugget) regularization strategy provides a provably robust alternative to Dropout for ANNs. We introduce a functional generalization of GPs and show that pointwise GP/RKHS error estimates lead to probabilistic and deterministic generalization error estimates for ResNets. When performed with feature maps, the proposed analysis identifies the (EPDiff) mean fields limit of trained ResNet parameters as the number of data points goes to infinity. The search for good architectures can be reduced to that of good kernels, and we show that the composition of warping regression blocks with reduced equivariant multichannel kernels (introduced here) recovers and generalizes CNNs to arbitrary spaces and groups of transformations.

Limiting Dynamics for Q-Learning with Memory One in Symmetric Two-Player, Two-Action Games

We develop a method based on computer algebra systems to represent the mutual pure strategy best-response dynamics of symmetric two-player, two-action repeated games played by players with a one-period memory. We apply this method to the iterated prisoner’s dilemma, stag hunt, and hawk-dove games and identify all possible equilibrium strategy pairs and the conditions for their existence. The only equilibrium strategy pair that is possible in all three games is the win-stay, lose-shift strategy. Lastly, we show that the mutual best-response dynamics are realized by a sample batch Q-learning algorithm in the infinite batch size limit.

Brownian bees in the infinite swarm limit

The Brownian bees model is a branching particle system with spatial selection. It is a system of N particles which move as independent Brownian motions in Rd and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that, as N→∞, the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper (Trans. Amer. Math. Soc. 374 (2021) 6269–6329)), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds; that is, that as N→∞, the equilibrium density of the particle system converges to the steady-state solution of the free boundary problem.

On Maximally Mixed Equilibria of Two-Dimensional Perfect Fluids

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution $\omega_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of $\omega_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy $\mathsf{E}_0$ corresponding to $\omega_0$. The set of said vorticity fields, denoted by $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit.

Characteristic power series of graph limits

In this note, we show how to obtain a “characteristic power series” of graphons – infinite limits of dense graphs – as the limit of normalized reciprocal characteristic polynomials. This leads to a new characterization of graph quasi-randomness and another perspective on spectral theory for graphons, a complete description of the function in terms of the spectrum of the graphon as a self-adjoint kernel operator. Interestingly, while we apply a standard regularization to classical determinants, it is unclear how necessary this is.

Homotopies in Multiway (Nondeterministic) Rewriting Systems as n-Fold Categories

We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order n may naturally be formalized as an n-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ∞-groupoid. Via Grothendieck’s homotopy hypothesis, this ∞-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of homotopical multiway systems may potentially be used for making formal connections to homotopy spaces upon which models relevant to physics may be instantiated.

Usefulness of higher-order system-size correction for macromolecule diffusion coefficients: A molecular dynamics study

Yeh and Hummer’s simplified estimation method has often been adopted to obtain diffusion coefficients for solute molecules using molecular dynamic simulation. However, the simplified formula is not necessarily valid when a small basic cell is used. Therefore, we conducted molecular dynamics simulations of aqueous protein solution to estimate the diffusion coefficient for the infinite dilution limit. We confirmed a deviation from the simplified formula in the simulation data and rationalized the discrepancy based on the unsimplified formula.

Dirac fermions with plaquette interactions. II. SU(4) phase diagram with Gross-Neveu criticality and quantum spin liquid

At sufficiently low temperatures, interacting electron systems tend to develop orders. Exceptions are quantum critical point (QCP) and quantum spin liquid (QSL), where fluctuations prevent the highly entangled quantum matter to an ordered state down to the lowest temperature. While the ramification of these states may have appeared in high-temperature superconductors, ultra-cold atoms, frustrated magnets and quantum Moir\'e materials, their unbiased presence remain elusive in microscopic two-dimensional lattice models. Here, we show by means of large-scale quantum Monte Carlo simulations of correlated electrons on the $\pi$-flux square lattice subjected to extended Hubbard interaction, that a Gross-Neveu QCP separating massless Dirac fermions and an columnar valence bond solid at finite interaction, and a possible Dirac QSL at the infinite yet tractable interaction limit emerge in a coherent sequence. These unexpected novel quantum states resides in this simple-looking model, unifying ingredients including emergent symmetry, deconfined fractionalization and the dynamic coupling between emergent matter and gauge fields, will have profound implications both in quantum many-body theory and understanding of the aforementioned experimental systems.

Pion and Kaon Distribution Amplitudes from Lattice QCD.

We present a state-of-the-art lattice QCD calculation of the pion and kaon light-cone distribution amplitudes (DAs) using large-momentum effective theory. The calculation is done at three lattice spacings a≈{0.06,0.09,0.12} fm and physical pion and kaon masses, with the meson momenta P_{z}={1.29,1.72,2.15} GeV. The result is nonperturbatively renormalized in a recently proposed hybrid scheme with self-renormalization, and extrapolated reliably to the continuum as well as the infinite momentum limit. We find a significant deviation of the pion and kaon DAs from the asymptotic form, and a large SU(3) flavor breaking effect in the kaon DA.

Celestial Yang-Mills amplitudes and D = 4 conformal blocks

We discuss the properties of recently constructed “single-valued” celestial four-gluon amplitudes. We show that the amplitude factorizes into the “current” part and the “scalar” part. The current factor is given by the group-dependent part of the Wess-Zumino-Witten correlator of four holomorphic currents with a non-vanishing level of Kač-Moody algebra. The scalar factor can be expressed in terms of a complex integral of the Koba-Nielsen form, similar to the integrals describing four-point correlators in Coulomb gas models and, more generally, in the infinite central charge limit of Liouville theory. The scalar part can be also obtained by a dimensional reduction of a single D = 4 conformal block and the shadow block from Minkowski space to the celestial sphere.

Elliptic K3 surfaces at infinite complex structure and their refined Kulikov models

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra.

Degenerated Calabi–Yau varieties with infinite components, moduli compactifications, and limit toroidal structures

For any degenerating Calabi–Yau family, we introduce a new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi–Yau varieties, parametrized by the rational points of the Kontsevich–Soibelman’s essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications (Sect. 3), locally modelled on limit toric varieties (Sect. 2.4), the way to attach a tropicalized family to a given Calabi–Yau family (Sect. 4), which are weakly related to each other.

The next-to-leading order Higgs impact factor in the infinite top-mass limit

We calculate the next-to-leading order correction to the impact factor (vertex) for the production of a forward Higgs boson, obtained in the infinite top-mass limit. We present the result both in the momentum representation and as superposition of the eigenfunctions of the leading-order BFKL kernel. This impact factor is a necessary ingredient for the description of the inclusive hadroproduction of a forward Higgs in the limit of small Bjorken x, as well as for the study of inclusive forward emissions of a Higgs boson in association with a backward identified object.

Bayesian Optimisation vs. Input Uncertainty Reduction

Simulators often require calibration inputs estimated from real-world data, and the estimate can significantly affect simulation output. Particularly when performing simulation optimisation to find an optimal solution, the uncertainty in the inputs significantly affects the quality of the found solution. One remedy is to search for the solution that has the best performance on average over the uncertain range of inputs yielding an optimal compromise solution. We consider the more general setting where a user may choose between either running simulations or querying an external data source, improving the input estimate and enabling the search for a more targeted, less compromised solution. We explicitly examine the trade-off between simulation and real data collection to find the optimal solution of the simulator with the true inputs. Using a value of information procedure, we propose a novel unified simulation optimisation procedure called Bayesian Information Collection and Optimisation that, in each iteration, automatically determines which of the two actions (running simulations or data collection) is more beneficial. We theoretically prove convergence in the infinite budget limit and perform numerical experiments demonstrating that the proposed algorithm is able to automatically determine an appropriate balance between optimisation and data collection.