Non-uniqueness phase in hyperbolic marked random connection models using the spherical transform

In this work we calculate in two different ways a well-known two-point two-loop massless Feynman diagram to illustrate the feasibility of “gluing” – that is, inserting or embedding – simpler diagrams into more complex ones, to allow for stepwise calculation in the context of negative dimensional integration method (NDIM). The two-loop diagram in question can be “glued” in two different ways and we show that both yield the same result and reproduce the one calculated via NDIM for the complete diagram, which, of course, is equivalent to the exact solution obtained by normal positive dimensional calculation. Furthermore, in the process we conclude that the usual massless off-shell triangle diagram result is not appropriate for the gluing purpose and present a new solution for it with only three hypergeometric functions F 4 that suits the gluing loop-by-loop technique.

It has long been known that two-point functions of conformal field theory (CFT) are nothing but the integral kernels of intertwining operators for two equivalent representations of conformal algebra. Such intertwining operators are known to fulfill some operator identities — the intertwining relations — in the representation space of conformal algebra. Meanwhile, it has been known that the S-matrix operator in scattering theory is nothing but the intertwining operator between the Hilbert spaces of in- and out-particles. Inspired by this algebraic resemblance, in this paper, we develop a simple Lie-algebraic approach to momentum-space two-point functions of thermal CFT living on the hyperbolic space–time [Formula: see text] by exploiting the idea of Kerimov’s intertwining operator approach to exact S-matrix. We show that in thermal CFT on [Formula: see text], the intertwining relations reduce to certain linear recurrence relations for two-point functions in the complex momentum space. By solving these recurrence relations, we obtain the momentum-space representations of advanced and retarded two-point functions as well as positive- and negative-frequency two-point Wightman functions for a scalar primary operator in arbitrary space–time dimension [Formula: see text].

Correlation functions in the multiple Ising model coupled to gravity

In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.

We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram \(\nabla _{p_c}\) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram \(\nabla _p\) is unbounded but diverges slowly as \(p \uparrow p_c\), as is expected to occur in percolation on \({\mathbb {Z}}^d\) at the upper-critical dimension \(d=6\). Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as \(p \uparrow p_c\) then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.

Considering the prevalence of the power-law distribution in user-item networks, hyperbolic space has attracted considerable attention and achieved impressive performance in the recommender system recently. The advantage of hyperbolic recommendation lies in that its exponentially increasing capacity is well-suited to describe the power-law distributed user-item network whereas the Euclidean equivalent is deficient. Nonetheless, it remains unclear which kinds of items can be effectively recommended by the hyperbolic model and which cannot. To address the above concerns, we take the most basic recommendation technique, collaborative filtering, as a medium, to investigate the behaviors of hyperbolic and Euclidean recommendation models. The results reveal that (1) tail items get more emphasis in hyperbolic space than that in Euclidean space, but there is still ample room for improvement; (2) head items receive modest attention in hyperbolic space, which could be considerably improved; (3) and nonetheless, the hyperbolic models show more competitive performance than Euclidean models. Driven by the above observations, we design a novel learning method, named hyperbolic informative collaborative filtering (HICF), aiming to compensate for the recommendation effectiveness of the head item while at the same time improving the performance of the tail item. The main idea is to adapt the hyperbolic margin ranking learning, making its pull and push procedure geometric-aware, and providing informative guidance for the learning of both head and tail items. Extensive experiments back up the analytic findings and also show the effectiveness of the proposed method. The work is valuable for personalized recommendations since it reveals that the hyperbolic space facilitates modeling the tail item, which often represents user-customized preferences or new products.

From previous work arXiv:2010.09811, the semiclassical backreaction equation in 1+1 dimensions was solved and a criterion was implemented to assess the validity of the semiclassical approximation in this case. The criterion involves the behavior of solutions to the linear response equation which describes perturbations about solutions to the semiclassical backreaction equation. The linear response equation involves a time integral over a two-point correlation function for the current induced by the quantum field and it is expected that significant growth in this two-point function (and therefore in quantum fluctuations) will result in significant growth in solutions to the linear response equation. It was conjectured for early times that the difference of two nearby solutions to the semiclassical backreaction equation, with similar initial conditions, can act as an approximate solution to the linear response equation. A comparative analysis between the approximate and numerical solutions to the linear response equation, for the critical scale for particle production, will be presented for the case of a massive, quantized spin 1/2 field in order to determine how robust the approximation method is for representing its solutions.

We demonstrate that the restoration of chiral symmetry at finite-T in a class of confining Dyson-Schwinger equation (DSE) models of QCD is a mean field transition, and that an accurate determination of the critical exponents using the chiral and thermal susceptibilities requires very small values of the current-quark mass: log_{10}(m/m_u) < -5. Other classes of DSE models characterised by qualitatively different interactions also exhibit a mean field transition. Incipient in this observation is the suggestion that mean field exponents are a result of the gap equation's fermion substructure and not of the interaction.

A non-critical mixture of polyethylene glycol (PEG600, M = 600 g mol−1) and polypropylene glycol (PPG1000, M = 1000 g mol−1) was investigated by ultrasonic and light scattering experiments in the one-phase region. The mass fraction of polypropylene glycol in the non-critical mixture was yPPG = 0.365. The explored temperature and frequency range of the ultrasonic experiment was 0.1 ≤ T − TP ≤ 17.3 K and 0.4 MHz ≤ f ≤ 30 MHz (TP: phase separation temperature). The frequency dependence of the ultrasonic attenuation of the non-critical mixture shows a relaxation behaviour typical for composition fluctuations. The data can be analysed by the dynamic scaling theory of Bhattacharjee and Ferrell which was formally extended to the non-critical case using the concept of a pseudospinodal temperature. The characteristic time scale of the concentration fluctuations is described by a frequency ωD = 2D/ξ2 where D is the mutual diffusion coefficient and ξ is the correlation length. In the frame of the pseudospinodal concept the temperature dependence of the characteristic frequency is expressed by ωD = ω0εzν with ε = (T − TPS)/TPS (TPS: pseudospinodal temperature, ε: reduced temperature, ω0: critical amplitude, zν: critical exponent). The temperature dependence of the frequency ωD was determined by the ultrasonic spectra. From this data, using mean field exponents, a value of ω0 = 9.4 MHz was estimated which is comparable to that of the critical mixture. The description of the ultrasonic data with mean field exponents is justified by the results of dynamic light scattering.

Deep hyperbolic convolutional model for knowledge graph embedding

Label inventories for fine-grained entity typing have grown in size and complexity. Nonetheless, they exhibit a hierarchical structure. Hyperbolic spaces offer a mathematically appealing approach for learning hierarchical representations of symbolic data. However, it is not clear how to integrate hyperbolic components into downstream tasks. This is the first work that proposes a fully hyperbolic model for multi-class multi-label classification, which performs all operations in hyperbolic space. We evaluate the proposed model on two challenging datasets and compare to different baselines that operate under Euclidean assumptions. Our hyperbolic model infers the latent hierarchy from the class distribution, captures implicit hyponymic relations in the inventory, and shows performance on par with state-of-the-art methods on fine-grained classification with remarkable reduction of the parameter size. A thorough analysis sheds light on the impact of each component in the final prediction and showcases its ease of integration with Euclidean layers.

Hyperbolic models are known to produce networks with properties observed empirically in most network datasets, including heavy-tailed degree distribution, high clustering, and hierarchical structures. As a result, several embedding algorithms have been proposed to invert these models and assign hyperbolic coordinates to network data. Current algorithms for finding these coordinates, however, do not quantify uncertainty in the inferred coordinates. We present BIGUE, a Markov chain Monte Carlo (MCMC) algorithm that samples the posterior distribution of a Bayesian hyperbolic random graph model. We show that the samples are consistent with current algorithms while providing added credible intervals for the coordinates and all network properties. We also show that some networks admit two or more plausible embeddings, a feature that an optimization algorithm can easily overlook.

We use the finiteness of the triangle diagram in order to establish that certain critical exponents take on their mean-field values. We again rely on the differential inequalities developed in chapter 3, and complement them with a differential inequality involving the triangle diagram. We then prove that, under the triangle condition, the critical exponents \(\delta \) and \(\beta \) take on their mean-field values \(\delta \) = 2 and \(\beta \) = 1.

We consider a neutral self-interacting massive scalar field defined in a d-dimensional Euclidean space. Assuming thermal equilibrium, we discuss the one-loop perturbative renormalization of this theory in the presence of rigid boundary surfaces (two parallel hyperplanes), which break translational symmetry. In order to identify the singular parts of the one-loop two-point and four-point Schwinger functions, we use a combination of dimensional and zeta-function analytic regularization procedures. The infinities which occur in both the regularized one-loop two-point and four-point Schwinger functions fall into two distinct classes: local divergences that could be renormalized with the introduction of the usual bulk counterterms, and surface divergences that demand counterterms concentrated on the boundaries. We present the detailed form of the surface divergences and discuss different strategies that one can assume to solve the problem of the surface divergences. We also briefly mention how to overcome the difficulties generated by infrared divergences in the case of Neumann–Neumann boundary conditions.

We consider the (λ/4!)(φ14+φ24) model on a d-dimensional Euclidean space, where all but one of the coordinates are unbounded. Translation invariance along the bounded coordinate, z, which lies in the interval [0,L], is broken because of the boundary conditions (BCs) chosen for the hyperplanes z=0 and z=L: DD and NN, where D denotes Dirichlet and N Neumann, respectively. The renormalization procedure up to one-loop order in the two-point function is applied, obtaining two main results. The first is the fact that the renormalization program requires the introduction of counterterms which are surface interactions. The second one is that the tadpole graphs for DD and NN have the same z dependent part in modulus but with opposite signs. We investigate the relevance of this fact to the elimination of surface divergences.