Mathematical Methods of Subjective Modeling in Scientific Research: I. The Mathematical and Empirical Basis

mathematical formalism for subjective modeling, based on modelling of uncertainty, reflecting unreliability of subjective information and fuzziness that is common for its content. The model of subjective judgments on values of an unknown parameter x ∈ X of the model M(x) of a research object is defined by the researcher–modeler as a space¹ (X, p(X), \(P{I^{\bar x}}\), \(Be{l^{\bar x}}\)) with plausibility\(P{I^{\bar x}}\) and believability \(Be{l^{\bar x}}\) measures, where x is an uncertain element taking values in X that models researcher—modeler’s uncertain propositions about an unknown x ∈ X, measures \(P{I^{\bar x}}\), \(Be{l^{\bar x}}\) model modalities of a researcher–modeler’s subjective judgments on the validity of each x ∈ X: the value of \(P{I^{\bar x}}(\tilde x = x)\) determines how relatively plausible, in his opinion, the equality \((\tilde x = x)\) is, while the value of \(Be{l^{\bar x}}(\tilde x = x)\) determines how the inequality \((\tilde x = x)\) should be relatively believed in. Versions of plausibility Pl and believability Bel measures and pl- and bel-integrals that inherit some traits of probabilities, psychophysics and take into account interests of researcher–modeler groups are considered. It is shown that the mathematical formalism of subjective modeling, unlike “standard” mathematical modeling, ?enables a researcher–modeler to model both precise formalized knowledge and non-formalized unreliable knowledge, from complete ignorance to precise knowledge of the model of a research object, to calculate relative plausibilities and believabilities of any features of a research object that are specified by its subjective model \(M(\tilde x)\), and if the data on observations of a research object is available, then it: ?enables him to estimate the adequacy of subjective model to the research objective, to correct it by combining subjective ideas and the observation data after testing their consistency, and, finally, to empirically recover the model of a research object.     

Likelihood analysis of the pMSSM11 in light of LHC 13-TeV data

We use MasterCode to perform a frequentist analysis of the constraints on a phenomenological MSSM model with 11 parameters, the pMSSM11, including constraints from \(\sim 36\)/fb of LHC data at 13 TeV and PICO, XENON1T and PandaX-II searches for dark matter scattering, as well as previous accelerator and astrophysical measurements, presenting fits both with and without the \((g-2)_\mu \) constraint. The pMSSM11 is specified by the following parameters: 3 gaugino masses \(M_{1,2,3}\), a common mass for the first-and second-generation squarks \(m_{\tilde{q}}\) and a distinct third-generation squark mass \(m_{\tilde{q}_3}\), a common mass for the first-and second-generation sleptons \(m_{\tilde{\ell }}\) and a distinct third-generation slepton mass \(m_{\tilde{\tau }}\), a common trilinear mixing parameter A, the Higgs mixing parameter \(\mu \), the pseudoscalar Higgs mass \(M_A\) and \(\tan \beta \). In the fit including \((g-2)_\mu \), a Bino-like \(\tilde{\chi }^0_{1}\) is preferred, whereas a Higgsino-like \(\tilde{\chi }^0_{1}\) is mildly favoured when the \((g-2)_\mu \) constraint is dropped. We identify the mechanisms that operate in different regions of the pMSSM11 parameter space to bring the relic density of the lightest neutralino, \(\tilde{\chi }^0_{1}\), into the range indicated by cosmological data. In the fit including \((g-2)_\mu \), coannihilations with \(\tilde{\chi }^0_{2}\) and the Wino-like \(\tilde{\chi }^\pm _{1}\) or with nearly-degenerate first- and second-generation sleptons are active, whereas coannihilations with the \(\tilde{\chi }^0_{2}\) and the Higgsino-like \(\tilde{\chi }^\pm _{1}\) or with first- and second-generation squarks may be important when the \((g-2)_\mu \) constraint is dropped. In the two cases, we present \(\chi ^2\) functions in two-dimensional mass planes as well as their one-dimensional profile projections and best-fit spectra. Prospects remain for discovering strongly-interacting sparticles at the LHC, in both the scenarios with and without the \((g-2)_\mu \) constraint, as well as for discovering electroweakly-interacting sparticles at a future linear \(e^+ e^-\) collider such as the ILC or CLIC.     

The Fermi–Pasta–Ulam Problem and Its Underlying Integrable Dynamics: An Approach Through Lyapunov Exponents

FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual \(\beta \)-model, perturbations of Toda include the usual \(\alpha +\beta \) model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent \(\chi \). More precisely, we consider statistically typical trajectories and study the asymptotics of \(\chi \) for large N (the number of particles) and small \(\varepsilon \) (the specific energy E / N), and find, for all models, asymptotic power laws \(\chi \simeq C\varepsilon ^a\), C and a depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of \(\chi \) introduced by Casetti, Livi and Pettini, originally formulated for the \(\beta \)-model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda.     

Chern–Simons,Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g., in the Chern–Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as \(p= \langle F, F\rangle \) where F is the curvature 2-form and \(\langle \cdot , \cdot \rangle \) is an invariant scalar product on the corresponding Lie algebra \(\mathfrak g\). The descent for p gives rise to an element \(\omega =\omega _3+\omega _2+\omega _1+\omega _0\) of mixed degree. The 3-form part \(\omega _3\) is the Chern–Simons form. The 2-form part \(\omega _2\) is known as the Wess–Zumino action in physics. The 1-form component \(\omega _1\) is related to the canonical central extension of the loop group LG. In this paper, we give a new interpretation of the low degree components \(\omega _1\) and \(\omega _0\). Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara–Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class p. In more detail, we define a 1-cocycle C which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara–Vergne equation F is mapped to \(\omega _1=C(F)\). Furthermore, the component \(\omega _0\) is related to the associator \(\Phi \) corresponding to F. It is surprising that while F and \(\Phi \) satisfy the highly nonlinear twist and pentagon equations, the elements \(\omega _1\) and \(\omega _0\) solve the linear descent equation.     

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields \(\psi \) and \(\phi \) defined on a countable set Z. For instance, \(Z=\mathbb {Z}^d\) or \(Z=\mathbb {Z}^d\times I\), where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of \(\psi \) and \(\phi \) enjoy a certain decay property, then all joint cumulants between \(\psi \) and \(\phi \) are \(\ell _2\)-summable in the precise sense described in the text. The decay assumption for the cumulants of \(\psi \) and \(\phi \) is a restricted \( \ell _1\) summability condition called \(\ell _1\)-clustering property. One immediate application of the results is given by a stochastic process \(\psi _t(x)\) whose state is \(\ell _1\)-clustering at any time t: then the above estimates can be applied with \(\psi =\psi _t\) and \(\phi =\psi _0\) and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any \(\ell _1\)-clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green–Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants     

Leptoquark mechanism of neutrino masses within the grand unification framework

We demonstrate the viability of the one-loop neutrino mass mechanism within the framework of grand unification when the loop particles comprise scalar leptoquarks (LQs) and quarks of the matching electric charge. This mechanism can be implemented in both supersymmetric and non-supersymmetric models and requires the presence of at least one LQ pair. The appropriate pairs for the neutrino mass generation via the up-type and down-type quark loops are \(S_3\)–\(R_2\) and \(S_{1,\,3}\)–\(\tilde{R}_2\), respectively. We consider two distinct regimes for the LQ masses in our analysis. The first regime calls for very heavy LQs in the loop. It can be naturally realized with the \(S_{1,\,3}\)–\(\tilde{R}_2\) scenarios when the LQ masses are roughly between \(10^{12}\) and \(5 \times 10^{13}\) GeV. These lower and upper bounds originate from experimental limits on partial proton decay lifetimes and perturbativity constraints, respectively. Second regime corresponds to the collider accessible LQs in the neutrino mass loop. That option is viable for the \(S_3\)–\(\tilde{R}_2\) scenario in the models of unification that we discuss. If one furthermore assumes the presence of the type II see-saw mechanism there is an additional contribution from the \(S_3\)–\(R_2\) scenario that needs to be taken into account beside the type II see-saw contribution itself. We provide a complete list of renormalizable operators that yield necessary mixing of all aforementioned LQ pairs using the language of SU(5). We furthermore discuss several possible embeddings of this mechanism in SU(5) and SO(10) gauge groups.     

Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$-term

We study D-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss–Bonnet term and the cosmological term \(\Lambda \). We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters \(H >0\) and h, corresponding to factor spaces of dimensions \(m >2\) and \(l > 2\), respectively. These solutions contain a fine-tuned \(\Lambda = \Lambda (x, m, l, \alpha )\), which depends upon the ratio \(h/H = x\), dimensions of factor spaces m and l, and the ratio \(\alpha = \alpha _2/\alpha _1\) of two constants (\(\alpha _2\) and \(\alpha _1\)) of the model. The master equation \(\Lambda (x, m, l,\alpha ) = \Lambda \) is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. The explicit solution for \(m = l\) is presented in “Appendix”. Imposing certain restrictions on x, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. We also consider a subclass of solutions with small enough variation of the effective gravitational constant G and show the stability of all solutions from this subclass.     

Logarithmic Finite-Size Correction in Non-neutral Two-Component Plasma on Sphere

We consider a general two-component plasma of classical pointlike charges \(+e\) (e is say the elementary charge) and \(-Z e\) (valency \(Z=1,2,\ldots \)), living on the surface of a sphere of radius R. The system is in thermal equilibrium at the inverse temperature \(\beta \), in the stability region against collapse of oppositely charged particle pairs \(\beta e^2 < 2/Z\). We study the effect of the system excess charge Qe on the finite-size expansion of the (dimensionless) grand potential \(\beta \varOmega \). By combining the stereographic projection of the sphere onto an infinite plane, the linear response theory and the planar results for the second moments of the species density correlation functions we show that for any \(\beta e^2 < 2/Z\) the large-R expansion of the grand potential is of the form \(\beta \varOmega \sim A_V R^2 + \left[ \chi /6 - \beta (Qe)^2/2\right] \ln R\), where \(A_V\) is the non-universal coefficient of the volume (bulk) part and the Euler number of the sphere \(\chi =2\). The same formula, containing also a non-universal surface term proportional to R, was obtained previously for the disc domain (\(\chi =1\)), in the case of the symmetric \((Z=1)\) two-component plasma at the collapse point \(\beta e^2=2\) and the jellium model \((Z\rightarrow 0)\) of identical e-charges in a fixed neutralizing background charge density at any coupling \(\beta e^2\) being an even integer. Our result thus indicates that the prefactor to the logarithmic finite-size expansion does not depend on the composition of the Coulomb fluid and its non-universal part \(-\beta (Qe)^2/2\) is independent of the geometry of the confining domain.     

A new decay mode of higher charmonium

We calculate the \(\Lambda _c{\bar{\Lambda }}_c\) partial decay width of the excited vector charmonium states around 4.6 GeV with the quark pair creation model. We find that the partial decay width of the \(\Lambda _c{\bar{\Lambda }}_c\) mode can reach up to several MeV for \(\psi (4S,~5S,~6S)\). In contrast, the partial \(\Lambda _c{\bar{\Lambda }}_c\) decay width of the states \(\psi (3D,~4D,~5D)\) is less than one MeV. If the enhancement Y(4630) reported by the Belle Collaboration in \(\Lambda _c{\bar{\Lambda }}_c\) invariant-mass distribution is the same structure as Y(4660), the Y(4660) resonance is most likely to be a S-wave charmonium state.     

Analysis of charmless two-body <Emphasis Type="Italic">B</Emphasis> decays in factorization-assisted topological-amplitude approach

We analyze charmless two-body non-leptonic B decays \(B \rightarrow PP, PV\) under the framework of a factorization-assisted topological-amplitude approach, where P(V) denotes a light pseudoscalar (vector) meson. Compared with the conventional flavor diagram approach, we consider the flavor SU(3) breaking effect assisted by a factorization hypothesis for topological diagram amplitudes of different decay modes, factorizing out the corresponding decay constants and form factors. The non-perturbative parameters of topology diagram magnitudes \(\chi \) and the strong phase \(\phi \) are universal; they can be extracted by \(\chi ^2\) fit from current abundant experimental data of charmless Bdecays. The number of free parameters and the \(\chi ^2\) per degree of freedom are both reduced compared with previous analyses. With these best fitted parameters, we predict branching fractions and CP asymmetry parameters of nearly 100 \(B_{u,d}\) and \(B_s\) decay modes. The long-standing \(\pi \pi \) and \(\pi K\)-CP puzzles are solved simultaneously.     

Phase Transition and Uniqueness of Levelset Percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function \(l:(0,\infty ) \rightarrow [0,\infty )\) to create the random field \(\Psi (y)=\sum _{x\in \eta }l(|x-y|),\) where \(\eta \) is a homogeneous Poisson process in \({\mathbb {R}}^d.\) The field \(\Psi \) is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded. In particular, we study the level sets \(\Psi _{\ge h}(y)\) containing the points \(y\in {\mathbb {R}}^d\) such that \(\Psi (y)\ge h.\) In the case where l has unbounded support, we give, for any \(d\ge 2,\) a necessary and sufficient condition on l for \(\Psi _{\ge h}(y)\) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is \(\Psi \) almost surely. Moreover, in this case and for \(d=2,\) we prove uniqueness of the infinite component of \(\Psi _{\ge h}\) when such exists, and we also show that the so-called percolation function is continuous below the critical value \(h_c\).     

The phenomenon of tristable stochastic resonance driven by $$\alpha $$-noise

In this paper, the tristable stochastic resonance (SR) phenomenon induced by \(\alpha \)-stable noise is analysed. The mechanism for realizing resonance is explored based on research concerning the potential function and resonant output of a system. The rules for resonance system parameters q, p, skewness parameter r and intensity amplification factor Q of \(\alpha \)-stable noise to act on the resonant output are explored under different values of stability index \(\alpha \) and asymmetric skewness \(\beta \) of \(\alpha \)-stable noise. The results will contribute to a reasonable selection of parameter-induced tristable SR system parameters under \(\alpha \)-stable noise, and lay the foundation for a practical engineering application of weak signal detection based on the SR.     

Lepton polarization asymmetries in rare semi-tauonic $$ b \rightarrow s $$ exclusive decays at FCC- ee

We consider measurements of exclusive rare semi-tauonic b-hadron decays, mediated by the \(b \rightarrow s \tau ^+ \tau ^-\) transition, at a future high-energy circular electron–positron collider (FCC-ee). We argue that the high boosts of b-hadrons originating from on-shell Z boson decays allow for a full reconstruction of the decay kinematics in hadronic \(\tau \) decay modes (up to discrete ambiguities). This, together with the potentially large statistics of \(Z\rightarrow b\bar{b}\), opens the door for the experimental determination of \(\tau \) polarizations in these rare b-hadron decays. In the light of the current experimental situation on lepton flavor universality in rare semileptonic B decays, we discuss the complementary short-distance physics information carried by the \(\tau \) polarizations and suggest suitable theoretically clean observables in the form of single- and double-\(\tau \) polarization asymmetries.     

Phase transition in anisotropic holographic superfluids with arbitrary dynamical critical exponent z and hyperscaling violation factor $$\alpha $$

Einstein-scalar-U(2) gauge field theory is considered in a spacetime characterized by \(\alpha \) and z, which are the hyperscaling violation factor and the dynamical critical exponent, respectively. We consider a dual fluid system of such a gravity theory characterized by temperature T and chemical potential \(\mu \). It turns out that there is a superfluid phase transition where a vector order parameter appears which breaks SO(3) global rotation symmetry of the dual fluid system when the chemical potential becomes a certain critical value. To study this system for arbitrary z and \(\alpha \), we first apply Sturm–Liouville theory and estimate the upper bounds of the critical values of the chemical potential. We also employ a numerical method in the ranges of \(1 \le z \le 4\) and \(0 \le \alpha \le 4\) to check if the Sturm–Liouville method correctly estimates the critical values of the chemical potential. It turns out that the two methods are agreed within 10 percent error ranges. Finally, we compute free energy density of the dual fluid by using its gravity dual and check if the system shows phase transition at the critical values of the chemical potential \(\mu _\mathrm{c}\) for the given parameter region of \(\alpha \) and z. Interestingly, it is observed that the anisotropic phase is more favored than the isotropic phase for relatively small values of z and \(\alpha \). However, for large values of z and \(\alpha \), the anisotropic phase is not favored.     

Excited state mass spectra of doubly heavy $$\Xi $$ baryons

In this paper, the mass spectra are obtained for doubly heavy \(\Xi \) baryons, namely, \(\Xi _{cc}^{+}\), \(\Xi _{cc}^{++}\), \(\Xi _{bb}^{-}\), \(\Xi _{bb}^{0}\), \(\Xi _{bc}^{0}\) and \(\Xi _{bc}^{+}\). These baryons consist of two heavy quarks (cc, bb, and bc) with a light (d or u) quark. The ground, radial, and orbital states are calculated in the framework of the hypercentral constituent quark model with Coulomb plus linear potential. Our results are also compared with other predictions, thus, the average possible range of excited states masses of these \(\Xi \) baryons can be determined. The study of the Regge trajectories is performed in (n, \(M^{2}\)) and (J, \(M^{2}\)) planes and their slopes and intercepts are also determined. Lastly, the ground state magnetic moments of these doubly heavy baryons are also calculated.     

Systematic study of rigid triaxiality in Ba–Pt nuclei and role of $$$$ subshell effect

A systematic variation of shape variables \(\beta \) and \(\gamma \) with N and \(N_\mathrm {p}N_\mathrm {n}\) is studied in the framework of an asymmetric rotor model of Davydov and Filippov for the \(Z=\) 50–82, \(N=\) 82–126 major shell space. The role of the \(Z=64\) subshell in producing smooth systematics has been discussed. The quadrupole moments are extracted after considering both axially symmetric and axially asymmetric nuclei. The correlation of \(\beta \) with \(\gamma \) together with the measured quadrupole moments indicates that \(\gamma \)-rigidity is better observed in nuclei with modest deformation.     

Scalar top quarks production in polarized photon-photon collisions at ILC

We study pair production of scalar top quarks (stop, \(\tilde t_1\)) in polarized photon-photon collisions with the subsequent decay of the top squarks into b quarks and charginos \(\tilde t_1 \to b\tilde \chi _1^ \pm\). We simulate this process by using PYTHIA6.4 for an electron beam energy \(2E_{beam}^e = \sqrt {s_{ee} } = 1000 GeV\). The energy spectrum of back-scattered photons is generated by CIRCE2 program. A set of criteria for physical variables is proposed which leads to a good separation of stop signal events from top quark pair production, being the main background. These criteria allow us to reconstruct the mass of the top squark, provided that the neutralino mass is known.     

Translation Invariant Extensions of Finite Volume Measures

We investigate the following questions: Given a measure \(\mu _\Lambda \) on configurations on a subset \(\Lambda \) of a lattice \(\mathbb {L}\), where a configuration is an element of \(\Omega ^\Lambda \) for some fixed set \(\Omega \), does there exist a measure \(\mu \) on configurations on all of \(\mathbb {L}\), invariant under some specified symmetry group of \(\mathbb {L}\), such that \(\mu _\Lambda \) is its marginal on configurations on \(\Lambda \)? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which \(\mathbb {L}=\mathbb {Z}^d\) and the symmetries are the translations. For the case in which \(\Lambda \) is an interval in \(\mathbb {Z}\) we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which \(\mathbb {L}\) is the Bethe lattice. On \(\mathbb {Z}\) we also consider extensions supported on periodic configurations, which are analyzed using de Bruijn graphs and which include the extensions with minimal entropy. When \(\Lambda \subset \mathbb {Z}\) is not an interval, or when \(\Lambda \subset \mathbb {Z}^d\) with \(d>1\), the LTI condition is necessary but not sufficient for extendibility. For \(\mathbb {Z}^d\) with \(d>1\), extendibility is in some sense undecidable.     

Gluino reach and mass extraction at the LHC in radiatively-driven natural SUSY

Radiatively-driven natural SUSY (RNS) models enjoy electroweak naturalness at the 10% level while respecting LHC sparticle and Higgs mass constraints. Gluino and top-squark masses can range up to several TeV (with other squarks even heavier) but a set of light Higgsinos are required with mass not too far above \(m_h\sim 125\) GeV. Within the RNS framework, gluinos dominantly decay via \(\tilde{g}\rightarrow t\tilde{t}_1^{*},\ \bar{t}\tilde{t}_1 \rightarrow t\bar{t}\widetilde{Z}_{1,2}\) or \(t\bar{b}\widetilde{W}_1^-+c.c.\), where the decay products of the higgsino-like \(\widetilde{W}_1\) and \(\widetilde{Z}_2\) are very soft. Gluino pair production is, therefore, signaled by events with up to four hard b-jets and large \(\not \!\!{E_T}\). We devise a set of cuts to isolate a relatively pure gluino sample at the (high-luminosity) LHC and show that in the RNS model with very heavy squarks, the gluino signal will be accessible for \(m_{\tilde{g}} < 2400 \ (2800)\) GeV for an integrated luminosity of 300 (3000) fb\(^{-1}\). We also show that the measurement of the rate of gluino events in the clean sample mentioned above allows for a determination of \(m_{\tilde{g}}\) with a statistical precision of 2–5% (depending on the integrated luminosity and the gluino mass) over the range of gluino masses where a 5\(\sigma \) discovery is possible at the LHC.     

Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots

Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues \(\{\lambda _1,\ldots ,\lambda _N\}\). We study the distribution of truncated linear statistics of the form \(\tilde{L}=\sum _{i=1}^p f(\lambda _i)\) with \(p<N\). This problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the “thinned ensembles” introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function for \(\tilde{L}\). Large deviations of linear statistics \(L=\sum _{i=1}^N f(\lambda _i)\) are usually dominated by the energy of the Coulomb gas, which scales as \(\sim N^2\), implying that the relative fluctuations are of order 1 / N. For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect. Because the entropy scales as \(\sim N\), the relative fluctuations are of order \(1/\sqrt{N}\). Our analysis relies on the mapping on a problem of p fictitious non-interacting fermions in N energy levels, which can exhibit both positive and negative effective (absolute) temperatures. We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here (\(f(\lambda )=1/\lambda \)), the corresponding phase diagram is separated in three different phases.