Renormalization Research Articles

Moderating effect of entrepreneurial self-efficacy on the strength of the mediated relationship between networking capabilities and SME performance via entrepreneurial competencies

The purpose of this study was determine the effect of networking capabilities on firm performance while controlling for entrepreneurial competencies, to determined the moderating effect of entrepreneurial self-efficacy on the relationship between entrepreneurial competencies and SME performance and ultimatrely to establish the moderating effect of entrepreneurial self-efficacy on the strength of the mediated relationship between networking capabilities and Small and Mediam Enterprises (SME) performance via entrepreneurial competencies. The study sdopted the positivist research philosophy asserting the existence of an objective reality that can be quantified, offering both explanatory and predictive capabilities. The research adopted an explanatory research design. Hayes Model 4 and 59 process Macro approach was used to analyse data and test the hypotheses. The results showed that networking capabilities had a significant direct effect on firm performance. The study also revealed that the interaction of entrepreneurial self- efficacy between the entrepreneurial competencies and firm performance was significant. Ultimately, the study determined the conditional indirect effect of Entrepreneurial Self-Efficacy (ESE) on the strength of the mediated relationship between networking capabilities and SME performance via entrepreneurial competencies. The study concluded that positive and significant moderated mediation analysis was evident. More importantly, the conditional indirect effect was much stronger with firms having lower levels of ESE. The study concluded that entrepreneurial self-efficacy has significant conditional indirect effect on the strength of the mediated relationship between networking capabilities and SME performance via entrepreneurial competencies. These findings emphasise the importance of self-efficacy, networking capabilities and competencies of the entrepreneurs as a predictor to improved firm performance.

Euclid preparation

We investigate the accuracy of the perturbative galaxy bias expansion in view of the forthcoming analysis of the Euclid spectroscopic galaxy samples. We compare the performance of a Eulerian galaxy bias expansion using state-of-the-art prescriptions from the effective field theory of large-scale structure (EFTofLSS) with a hybrid approach based on Lagrangian perturbation theory and high-resolution simulations. These models are benchmarked against comoving snapshots of the flagship I N-body simulation at z = (0.9, 1.2, 1.5, 1.8), which have been populated with Hα galaxies leading to catalogues of millions of objects within a volume of about 58 h−3 Gpc3. Our analysis suggests that both models can be used to provide a robust inference of the parameters (h, ωc) in the redshift range under consideration, with comparable constraining power. We additionally determine the range of validity of the EFTofLSS model in terms of scale cuts and model degrees of freedom. From these tests, it emerges that the standard third-order Eulerian bias expansion – which includes local and non-local bias parameters, a matter counter term, and a correction to the shot-noise contribution – can accurately describe the full shape of the real-space galaxy power spectrum up to the maximum wavenumber of kmax = 0.45 h Mpc−1, and with a measurement precision of well below the percentage level. Fixing either of the tidal bias parameters to physically motivated relations still leads to unbiased cosmological constraints, and helps in reducing the severity of projection effects due to the large dimensionality of the model. We finally show how we repeated our analysis assuming a volume that matches the expected footprint of Euclid, but without considering observational effects, such as purity and completeness, showing that we can get constraints on the combination (h, ωc) that are consistent with the fiducial values to better than the 68% confidence interval over this range of scales and redshifts.

Conformal perturbation theory for n-point functions: structure constant deformation

We consider conformal perturbation theory for n-point functions on the sphere in general 2D CFTs to first order in coupling constant. We regulate perturbation integrals using canonical hard disk excisions of size ϵ around the fixed operator insertions, and identify the full set of counter terms which are sufficient to regulate all such integrated n-point functions. We further explore the integrated 4-point function which computes changes to the structure constants of the theory. Using an sl(2) map, the three fixed locations of operators are mapped to 0, 1, and ∞. We show that approximating the mapped excised regions to leading order in ϵ does not lead to the same perturbative shift to the structure constant as the exact in ϵ region. We explicitly compute the correction back to the exact in ϵ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in ϵ region to obtain agreement with the exact results for structure constants in this theory.

A New 2D Formulation of Modified General Relativity

AbstractIt is well known that the Einstein–Hilbert action in two dimensions is topological and yields an identically vanishing Einstein tensor. Consequently one is faced with difficulties when formulating a non‐trivial gravity model. The authors present a new, intrinsically two‐dimensional, approach to this problem based on the Einstein action. This yields a well defined variational approach which results in new field equations that break diffeomorphism invariance. This proposed approach does not require the introduction of additional scalar fields, nor the use of conformal transformations. However, how including conformal counter terms leads to equivalent results is shown. In doing so, an explanation for why previous approaches worked can be provided. Solutions to the field equations are briefly discussed.

Parisi–Wu quantization, CP violation and beyond the standard model

The Parisi–Wu scheme of quantization opens up the possibility of using anomalous fermionic gauge theories. An analysis of ultra-violet divergences reveals that the structure of counter terms is different from what is expected in conventional quantization schemes. In this paper, it is argued that there exists a possible mechanism of CP violation that requires at least three generations of quarks, a result well known from a phenomenological analysis of mass mixing of quarks. A few observations on possible ways of going beyond the Standard Model are included.

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We develop Standard Model Effective Field Theory (SMEFT) predictions of σ(GG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{GG} $$\\end{document} → h), Γ(h → GG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{GG} $$\\end{document}), Γ(h → AA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{AA} $$\\end{document}) to incorporate full two loop Standard Model results at the amplitude level, in conjunction with dimension eight SMEFT corrections. We simultaneously report consistent Γ(h → Ψ¯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\Psi}\\Psi $$\\end{document}) results including leading QCD corrections and dimension eight SMEFT corrections. This extends the predictions of the former processes Γ, σ to a full set of corrections at Ov¯T2/Λ216π22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\overline{v}}_T^2/{\\varLambda}^2{\\left(16{\\pi}^2\\right)}^2\\right) $$\\end{document} and Ov¯T4/Λ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\overline{v}}_T^4/{\\Lambda}^4\\right) $$\\end{document}, where v¯T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\overline{v}}_T $$\\end{document} is the electroweak scale vacuum expectation value and Λ is the cut off scale of the SMEFT. Throughout, cross consistency between the operator and loop expansions is maintained by the use of the geometric SMEFT formalism. For Γ(h → Ψ¯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\Psi}\\Psi $$\\end{document}), we include results at Ov¯T2/Λ216π2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\overline{v}}_T^2/{\\Lambda}^2\\left(16{\\pi}^2\\right)\\right) $$\\end{document} in the limit where subleading mΨ → 0 corrections are neglected. We clarify how gauge invariant SMEFT renormalization counterterms combine with the Standard Model counter terms in higher order SMEFT calculations when the Background Field Method is used. We also update the prediction of the total Higgs width in the SMEFT to consistently include some of these higher order perturbative effects.

New diffusive interface model for pitting corrosion

In this paper, a diffusive interface model for pitting corrosion is proposed, which can be used to simulate diffusion- and activation-controlled pitting corrosion with the pit morphologies and the distribution of metal cation concentration. The diffusive interface equation is derived from the general interface advection-like equation coupled with the description of the normal velocity of the moving pitting boundary from the sharp interface model. The corrosion rate is expressed in an explicit way through the diffusion field in the diffusion-controlled regime, and the activation-controlled corrosion is expressed by introducing the interface kinetic parameter, which greatly simplifies the model. The curvature effect can be subtracted back out directly by introducing a counter term. Predicted results from numerical simulations are compared with the analytical solution to determine numerical parameter in one-dimensional pencil electrode test. The effect of local curvature on simulated semicircle corrosion is discussed. Then, the application scenarios of the model including electropolishing, single and multiple corrosion, corrosion with composite materials are presented.

Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system.

Theoretical examination of QED Hamiltonian in relativistic molecular orbital theory.

Theoretical discussions are given on issues in relativistic molecular orbital theory to which the quantum electrodynamics (QED) Hamiltonian is applied. First, several QED Hamiltonians previously proposed are sifted by the orbital rotation invariance, the charge conjugation and time reversal invariance, and the nonrelativistic limit. The discussion on orbital rotation invariance shows that orbitals giving a stationary point of total energy should be adopted for QED Hamiltonians that are not orbital rotation invariant. A new total energy expression is then proposed, in which a counter term corresponding to the energy of the polarized vacuum is subtracted from the total energy. This expression prevents the possibility of total energy divergence due to electron correlations, stemming from the fact that the QED Hamiltonian does not conserve the number of particles. Finally, based on the Hamiltonian and energy expression, the Dirac-Hartree-Fock (DHF) and electron correlation methods are reintroduced. The QED-based DHF equation is shown to give information on positrons from negative-energy orbitals while having the same form as the conventional DHF equation. Three electron correlation methods are derived: the QED-based configuration interactions and single- and multireference perturbation methods. Numerical calculations show that the total energy of the QED Hamiltonian indeed diverged and that the counter term is effective in avoiding the divergence. The relativistic molecular orbital theory presented in this article also provides a methodology for dealing with systems containing positrons based on the QED Hamiltonian.

Stochastic quantization and holographic Wilsonian renormalization group of scalar theory with generic mass, self-interaction and multiple trace deformation

We explore the mathematical relationship between holographic Wilsonian renormalization group (HWRG) and stochastic quantization (SQ) of scalar field theory with its generic mass, self-interaction and [Formula: see text]-multiple-trace deformation on the [Formula: see text]-dimensional conformal boundary defined in AdS[Formula: see text] space–time. We understand that once we define our Euclidean action, [Formula: see text] as [Formula: see text], then the stochastic process will reconstruct the HWRG data via solving Langevin equation and computing stochastic correlation functions. The [Formula: see text] is given by [Formula: see text], where [Formula: see text] is the boundary counter term and [Formula: see text] is the boundary deformation which gives a boundary condition. In our study, we choose the boundary condition adding (marginal)[Formula: see text]-multiple-trace deformation to the holographic dual field theory. In this theory, we establish maps between fictitious time, [Formula: see text] evolution of stochastic [Formula: see text]-point, ([Formula: see text])-point correlation functions and the (AdS)radial, [Formula: see text] evolution of [Formula: see text]-multiple-trace and ([Formula: see text])-multiple-trace deformations, respectively, once we take identifications of [Formula: see text] and between some of the constants appearing in both sides.

Effects of fiber content, inclination angle, and casting point on the fiber-matrix interaction of High-Strength Fiber-Reinforced Self Compacting Concrete

This experimental study examines the pullout behavior of aligned and inclined steel fibers embedded in High-Strength Fiber-Reinforced Self Compacting Concrete (HSFRSCC). The effects of four fiber contents (0, 0.25, 0.50, and 0.75%), three inclination angles (0°, 30°, and 45°), and three concrete casting points on the formwork were investigated. The analyses were performed based on load-slip relationships, maximum pullout load, dissipated energy, failure mode, and spalling area. The fiber pullout behavior evidenced the importance of the presence of fibers in the concrete matrix. The pullout loads and pullout energy presented a better performance than the unreinforced matrix when the fiber content was increased to 0.75%; the confinement effect of the fibers in the matrix explains this mechanism. Regarding the failure mode, some inclined fibers at 30° and 45° presented rupture during pullout, more recurrent for 45° and for the matrices that offered greater confinement (0 and 0.75%). Furthermore, the spalling area increased with the inclination angle; it was found that the presence of fibers in the matrix influences the propagation of cracks during the matrix spalling. In some cases, these can increase the susceptibility of fiber rupture during the pullout. Finally, it was inconclusive that the pulled fibers' position relative to the casting point and, consequently, the fiber orientation influences the pullout load.

Self-consistent thermodynamic potential for magnetized QCD matter

Within the two-flavor Nambu--Jona-Lasinio model, we derive a self-consistent thermodynamic potential $\Omega$ for a QCD matter in an external magnetic field $B$. To be consistent with Schwinger's renormalization spirit, counter terms with vacuum quark mass are introduced into $\Omega$ and then the explicit $B$-dependent parts can be regularized in a cutoff-free way. Following that, explicit expressions of gap equation and magnetization can be consistently obtained according to the standard thermodynamic relations. The formalism is able to reproduce the paramagnetic feature of a QCD matter without ambiguity. For more realistic study, a running coupling constant is also adopted to account for the inverse magnetic catalysis effect. It turns out that the running coupling would greatly suppress magnetization at large $B$ and is important to reproduce the temperature enhancement effect to magnetization. The case with finite baryon chemical potential is also explored: no sign of first-order transition is found by varying $B$ for the running coupling and the de Haas-van Alphen oscillation shows up in the small $B$ region.

The Representation of Humanity in Asif Currimbhoy’s Inquilab

In recent decades, mankind has been progressing towards the advancement in the name of technology. At the hand of advancement and development, the elements of basic humanity are doomed under the hypocrisy in society. There is a need for the insurgence of humanity over inhumanity to save mankind. Every human being, in the interaction of self and society, tries to accommodate the soul for the best of human nature. If this is the element to be drawn for the artistic play, Asif Currimbhoy has the advantage in every manner to make use of the elemental psychic drama molded into an excellent play. The present play by Asif Currimbhoy, Inquilab portrays human nature with the zeal of representation of humanity over the doomed system of injustice caused in society. It tries to pave a way to bear the torch towards humanity over adversity of the human psyche in the name of feudalism. Finally, it also searches for the redressal of mechanisms in the human behavior, as the Naxalite way of life is not a short cut, nor a sole way out for many filthy problems of society and does not bring the wanted means of democracy.

Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime

Quantum treatment of physical reference frame leads to the Ricci flow of quantum spacetime, which is a quite rigid framework to quantum and renormalization effect of gravity. The theory has a low characteristic energy scale described by a unique constant: the critical density of the universe. At low energy long distance (cosmic or galactic) scale, the theory modifies Einstein's gravity which naturally gives rise to a cosmological constant as a counter term of the Ricci flow at leading order and an effective scale dependent Einstein-Hilbert action. In the weak and static gravity limit, the framework gives rise to a transition trend away from Newtonian gravity and similar to the MOdified Newtonian Dynamics (MOND) around the characteristic scale. When local curvature is large, Newtonian gravity is recovered. When local curvature is low enough to be comparable with the asymptotic background curvature corresponding to the characteristic energy scale, the transition trend produces the baryonic Tully-Fisher relation. For intermediate general curvature around the background curvature, the interpolating Lagrangian function yields a similar transition trend to the observed radial acceleration relation of galaxies. When the baryonic matter density is much lower than the critical density at the outskirt of a galaxy, there may be a universal "acceleration floor" corresponding to the acceleration expansion of the universe, which differs from MOND at its deep-MOND limit. The critical acceleration constant $a_0$ introduced in MOND is related to the low characteristic energy scale of the theory. The cosmological constant gives a universal leading order contribution to it and the flow effect gives the next order scale dependent contribution, which equivalently induces the "cold dark matter" to the theory. $a_0$ is consistent with galaxian data when the "dark matter" is about 5 times the baryonic matter.

Cosmological structure formation and soliton phase transition in fuzzy dark matter with axion self-interactions

ABSTRACT We investigate cosmological structure formation in fuzzy dark matter (FDM) with the attractive self-interaction (SI) with numerical simulations. Such a SI would arise if the FDM boson were an ultra-light axion, which has a strong CP symmetry-breaking scale (decay constant). Although weak, the attractive SI may be strong enough to counteract the quantum ‘pressure’ and alter structure formation. We find in our simulations that the SI can enhance small-scale structure formation, and soliton cores above a critical mass undergo a phase transition, transforming from dilute to dense solitons.

Interferometric H i intensity mapping: perturbation theory predictions and foreground removal effects

ABSTRACT We provide perturbation theory predictions for the H i intensity mapping power spectrum multipoles using the Effective Field Theory of Large Scale Structure, which should allow us to exploit mildly non-linear scales. Assuming survey specifications typical of proposed interferometric H i intensity mapping experiments like Canadian Hydrogen Observatory and Radio transient Detector and PUMA, and realistic ranges of validity for the perturbation theory modelling, we run mock full shape Markov chain Monte Carlo (MCMC) analyses at z = 0.5, and compare with Stage-IV optical galaxy surveys. We include the impact of 21cm foreground removal using simulations-based prescriptions, and quantify the effects on the precision and accuracy of the parameter estimation. We vary 11 parameters in total: three cosmological parameters, seven bias and counter terms parameters, and the H i brightness temperature. Amongst them, the four parameters of interest are: the cold dark matter density, ωc, the Hubble parameter, h, the primordial amplitude of the power spectrum, As, and the linear H i bias, b1. For the best-case scenario, we obtain unbiased constraints on all parameters with $\lt 3{{\ \rm per\ cent}}$ errors at $68{{\ \rm per\ cent}}$ confidence level. When we include the foreground removal effects, the parameter estimation becomes strongly biased for ωc, h, and b1, while As is less biased (<2σ). We find that scale cuts $k_{\rm min} \ge 0.03 \ h\,\mathrm{Mpc}^{-1}$ are required to return accurate estimates for ωc and h, at the price of a decrease in the precision, while b1 remains strongly biased. We comment on the implications of these results for real data analyses.

Anyon scattering from lightcone Hamiltonian: the singlet channel

We study U(N) Chern-Simons theory coupled to massive fundamental fermions in the lightcone Hamiltonian formalism. Focusing on the planar limit, we introduce a consistent regularization scheme, identify the counter terms needed to restore relativistic invariance, and formulate scattering theory in terms of unambiguously defined asymptotic states. We determine the 2 → 2 planar S-matrix element in the singlet channel by solving the Lippmann-Schwinger equation to all orders, establishing a result previously conjectured in the literature.

NLO JIMWLK evolution with massive quarks

NLO evolution of the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation with massless quarks was derived a few years ago. We make a step further to compute the evolution kernels focusing on the effects due to finite quark masses. To this goal, the light-cone wave function of a fast moving dilute hadronic projectile is computed up to mathcal{O} (g3) in QCD coupling constant. Compared with the massless case, a new IR divergence emerges, which is eventually canceled by a mass dependent counter term. Our results extend the theoretical tools used in physics of gluon saturation and aim at improving precision in future phenomenological applications.

Cosmological structure formation in complex scalar field dark matter versus real ultralight axions: A comparative study using class

(abridged) We continue the study of $\Lambda$SFDM cosmologies, which differ from $\Lambda$CDM in that CDM is replaced by scalar field dark matter (SFDM) by calculating the evolution of the background Universe, as well as linear perturbations, focusing on scalar modes. We consider models with complex scalar field with a repulsive, quartic self-interaction (SI), and models without SI, referred to as fuzzy dark matter (FDM). To this end, we modify the Boltzmann code CLASS, to incorporate the physics of complex SFDM which has as one of its characteristics that its equation of state is maximally stiff in the very early Universe, dominating then over all the other cosmic components, even over radiation. We calculate CMB and matter power spectra as well as unconditional Press-Schechter halo mass functions for various models, expanding previous literature that were limited either to the background, or to a semi-analytical approach to SFDM density perturbations neglecting the early stiff phase. Comparing our results of each, SFDM and FDM, with real-field ultralight axion models (ULAs) without SI, we characterize the differences between the respective background evolution and linear structure growth. Our calculations confirm previous results of recent literature, implying that SFDM models with $\gtrsim$ kpc-size halo cores are disfavored, questioning their ability to explain the small-scale problems on dwarf-galactic scales. Also we find that the kinetic energy due to the phase of the complex field leads to marked differences between SFDM/FDM versus ULAs. The mild falloff in the SFDM power spectrum toward high k is similar to that of CDM but based on different effects, namely the rapidly shrinking Jeans mass for SFDM as opposed to the Meszaros effect for CDM. In addition, we find that the sharp cutoff in the ULA power spectrum is also followed by a mild falloff, albeit at very small power.

Large deviations analysis for random combinatorial partitions with counter terms

In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. The large deviation rate functions are normalised limiting free energies and the main focus is to study their minimiser for various Gibbsian ensembles with respect to the reference measure which is a probabilistic version of the ideal Bose gas. Scaling limits of similar models have been studied recently (Fatkullin and Slastikov 2018 arXiv:1801.00812v2; Fatkullin and Xue 2021 J. Stat. Phys. 183 22) going back to (Vershik 1996 Func. Anal. Appl. 30 90–105). After studying the reference model, we provide a complete analysis of two mean field models, one of which is well-know (Benfatto et al 2005 J. Math. Phys. 46 033303) and the other one is the cycle mean field model. Both models show critical behaviour despite their rate functions having unique minimiser. The main focus is then a model with negative counter term, the probabilistic version of the so-called Huang–Yang–Luttinger model (van den Berg et al 1988 Commun. Math. Phys. 118 61–85). Criticality in this model is the existence of a critical parameter for which two simultaneous minimiser exists. At criticality an order parameter is introduced as the double limits for the density of cycles with diverging length, and as such it extends recent work in (Adams and Dickson 2021 Ann. Henri Poincaré 22 1535–60).