Asymptotic Equivalence of Probability Measures and Stochastic Processes

Let \(P_n\) and \(Q_n\) be two probability measures representing two different probabilistic models of some system (e.g., an n-particle equilibrium system, a set of random graphs with n vertices, or a stochastic process evolving over a time n) and let \(M_n\) be a random variable representing a “macrostate” or “global observable” of that system. We provide sufficient conditions, based on the Radon–Nikodym derivative of \(P_n\) and \(Q_n\), for the set of typical values of \(M_n\) obtained relative to \(P_n\) to be the same as the set of typical values obtained relative to \(Q_n\) in the limit \(n\rightarrow \infty \). This extends to general probability measures and stochastic processes the well-known thermodynamic-limit equivalence of the microcanonical and canonical ensembles, related mathematically to the asymptotic equivalence of conditional and exponentially-tilted measures. In this more general sense, two probability measures that are asymptotically equivalent predict the same typical or macroscopic properties of the system they are meant to model.     

Determinations of $$|V_{cb}|$$ and $$|V_{ub}|$$|Vub| from baryonic $$\Lambda _b$$Λb decays

We present the first attempt to extract \(|V_{cb}|\) from the \(\Lambda _b\rightarrow \Lambda _c^+\ell \bar{\nu }_\ell \) decay without relying on \(|V_{ub}|\) inputs from the B meson decays. Meanwhile, the hadronic \(\Lambda _b\rightarrow \Lambda _c M_{(c)}\) decays with \(M=(\pi ^-,K^-)\) and \(M_c=(D^-,D^-_s)\) measured with high precisions are involved in the extraction. Explicitly, we find that \(|V_{cb}|=(44.6\pm 3.2)\times 10^{-3}\), agreeing with the value of \((42.11\pm 0.74)\times 10^{-3}\) from the inclusive \(B\rightarrow X_c\ell \bar{\nu }_\ell \) decays. Furthermore, based on the most recent ratio of \(|V_{ub}|/|V_{cb}|\) from the exclusive modes, we obtain \(|V_{ub}|=(4.3\pm 0.4)\times 10^{-3}\), which is close to the value of \((4.49\pm 0.24)\times 10^{-3}\) from the inclusive \(B\rightarrow X_u\ell \bar{\nu }_\ell \) decays. We conclude that our determinations of \(|V_{cb}|\) and \(|V_{ub}|\) favor the corresponding inclusive extractions in the B decays.     

Hyperscaling for Oriented Percolation in $$$$ Space–Time Dimensions

Consider nearest-neighbor oriented percolation in \(d+1\) space–time dimensions. Let \(\rho ,\eta ,\nu \) be the critical exponents for the survival probability up to time t, the expected number of vertices at time t connected from the space–time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality \(d\nu \ge \eta +2\rho \), which holds for all \(d\ge 1\) and is a strict inequality above the upper-critical dimension 4, becomes an equality for \(d=1\), i.e., \(\nu =\eta +2\rho \), provided existence of at least two among \(\rho ,\eta ,\nu \). The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. [6].     

Asymptotic behavior of the Schrödinger–Debye system with refractive index of quadratic wave amplitude

We obtain local well-posedness for the one-dimensional Schrödinger–Debye interactions in nonlinear optics in the spaces \(L^2\times L^p,\; 1\le p < \infty \). When \(p=1\) we show that the local solutions extend globally. In the focusing regime, we consider a family of solutions \(\{(u_{\tau }, v_{\tau })\}_{\tau >0}\) in \( H^1\times H^1\) associated to an initial data family \(\{(u_{\tau _0},v_{\tau _0})\}_{\tau >0}\) uniformly bounded in \(H^1\times L^2\), where \(\tau \) is a small response time parameter. We prove that \(\left( u_{\tau }, v_{\tau }\right) \) converges to \(\left( u, -|u|^2\right) \) in the space \(L^{\infty }_{[0, T]}L^2_x\times L^1_{[0, T]}L^2_x\) whenever \(u_{\tau _0}\) converges to \(u_0\) in \(H^1\) as long as \(\tau \) tends to 0, where u is the solution of the one-dimensional cubic nonlinear Schrödinger equation with the initial data \(u_0\). The convergence of \(v_{\tau }\) for \(-|u|^2\) in the space \(L^{\infty }_{[0, T]}L^2_x\) is shown under compatibility conditions of the initial data. For non-compatible data, we prove convergence except for a corrector term which looks like an initial layer phenomenon.     

Large Communities in a Scale-Free Network

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability \(p\in (0,1)\), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph \(G_{t}\) produced by this so-called generalized linear preferential (GLP) model at time t contains a complete subgraph whose vertex set cardinality is given by \(t^\alpha \), where \(\alpha = (1-\varepsilon )\frac{1-p}{2-p}\), for any small \(\varepsilon >0\) asymptotically almost surely.     

Kaonic production of $ \Lambda$(1405) off deuteron target in chiral dynamics

The K^--induced production of \( \Lambda\)(1405) is investigated in K ^- d \( \rightarrow\) \( \pi\) \( \Sigma\) n reactions based on coupled-channels chiral dynamics, in order to discuss the resonance position of the \( \Lambda\)(1405) in the \( \bar{{K}}\) N channel. We find that the K ^- d \( \rightarrow\) \( \Lambda\)(1405)n process favors the production of \( \Lambda\)(1405) initiated by the \( \bar{{K}}\) N channel. The present approach indicates that the \( \Lambda\)(1405) -resonance position is 1420MeV rather than 1405MeV in the \( \pi\) \( \Sigma\) invariant-mass spectra of K ^- d \( \rightarrow\) \( \pi\) \( \Sigma\) n reactions. This is consistent with an observed spectrum of the K ^- d \( \rightarrow\) \( \pi^{{+}}_{}\) \( \Sigma^{{-}}_{}\) n with 686-844MeV/c incident K^- by bubble chamber experiments done in the 70s. Our model also reproduces the measured \( \Lambda\)(1405) production cross-section.     

Critical Two-Point Function for Long-Range <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) Models Below the Upper Critical Dimension

We consider the n-component \(|\varphi |^4\) lattice spin model (\(n \ge 1\)) and the weakly self-avoiding walk (\(n=0\)) on \(\mathbb Z^d\), in dimensions \(d=1,2,3\). We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as \(r^{-(d+\alpha )}\) with \(\alpha \in (0,2)\). The upper critical dimension is \(d_c=2\alpha \). For \(\varepsilon >0\), and \(\alpha = \frac{1}{2} (d+\varepsilon )\), the dimension \(d=d_c-\varepsilon \) is below the upper critical dimension. For small \(\varepsilon \), weak coupling, and all integers \(n \ge 0\), we prove that the two-point function at the critical point decays with distance as \(r^{-(d-\alpha )}\). This “sticking” of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.     

The decays of $$\bar{B}^0$$, $$\bar{B}^0_s$$B¯s0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson

We investigate the decays of \(\bar{B}^0_s\), \(\bar{B}^0\) and \(B^-\) into \(\eta _c\) plus a scalar or vector meson in a theoretical framework by taking into account the dominant process for the weak decay of \(\bar{B}\) meson into \(\eta _c\) and a \(q\bar{q}\) pair. After hadronization of this \(q\bar{q}\) component into pairs of pseudoscalar mesons we obtain certain weights for the pseudoscalar meson-pseudoscalar meson components. In addition, the \(\bar{B}^0\) and \(\bar{B}^0_s\) decays into \(\eta _c\) and \(\rho ^0\), \(K^*\) are evaluated and compared to the \(\eta _c\) and \(\phi \) production. The calculation is based on the postulation that the scalar mesons \(f_0(500)\), \(f_0(980)\) and \(a_0(980)\) are dynamically generated states from the pseudoscalar meson-pseudoscalar meson interactions in S-wave. Up to a global normalization factor, the \(\pi \pi \), \(K \bar{K}\) and \(\pi \eta \) invariant mass distributions for the decays of \(\bar{B}^0_s \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0_s \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0 \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^0 \eta \), \(B^- \rightarrow \eta _c K^0 K^-\) and \(B^- \rightarrow \eta _c \pi ^- \eta \) are predicted. Comparison is made with the limited experimental information available and other theoretical calcualtions. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the nature of the low lying scalar mesons, \(f_0(500), f_0(980)\) and \(a_0(980)\).     

Lifshits Tails for Randomly Twisted Quantum Waveguides

We consider the Dirichlet Laplacian \(H_\gamma \) on a 3D twisted waveguide with random Anderson-type twisting \(\gamma \). We introduce the integrated density of states \(N_\gamma \) for the operator \(H_\gamma \), and investigate the Lifshits tails of \(N_\gamma \), i.e. the asymptotic behavior of \(N_\gamma (E)\) as \(E \downarrow \inf \mathrm{supp}\, dN_\gamma \). In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.     

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Let G be a connected graph in which almost all vertices have linear degrees and let \(\mathcal {T}\) be a uniform spanning tree of G. For any fixed rooted tree F of height r we compute the asymptotic density of vertices v for which the r-ball around v in \(\mathcal {T}\) is isomorphic to F. We deduce from this that if \(\{G_n\}\) is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of \(G_n\) locally converges to a multi-type branching process defined in terms of W. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least \(e^{-1}-\mathsf {o}(1)\), the density of vertices of degree 2 is at most \(e^{-1}+\mathsf {o}(1)\) and the density of vertices of degree \(k\geqslant 3\) is at most \({(k-2)^{k-2} \over (k-1)! e^{k-2}} + \mathsf {o}(1)\). These bounds are sharp.     

$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ decay in the aligned two-Higgs-doublet model

In the aligned two-Higgs-doublet model, we perform a complete one-loop computation of the short-distance Wilson coefficients \(C_{7,9,10}^{(\prime )}\), which are the most relevant ones for \(b\rightarrow s\ell ^+\ell ^-\) transitions. It is found that, when the model parameter \(\left| \varsigma _{u}\right| \) is much smaller than \(\left| \varsigma _{d}\right| \), the charged scalar contributes mainly to chirality-flipped \(C_{9,10}^\prime \), with the corresponding effects being proportional to \(\left| \varsigma _{d}\right| ^2\). Numerically, the charged-scalar effects fit into two categories: (A) \(C_{7,9,10}^\mathrm {H^\pm }\) are sizable, but \(C_{9,10}^{\prime \mathrm {H^\pm }}\simeq 0\), corresponding to the (large \(\left| \varsigma _{u}\right| \), small \(\left| \varsigma _{d}\right| \)) region; (B) \(C_7^\mathrm {H^\pm }\) and \(C_{9,10}^{\prime \mathrm {H^\pm }}\) are sizable, but \(C_{9,10}^\mathrm {H^\pm }\simeq 0\), corresponding to the (small \(\left| \varsigma _{u}\right| \), large \(\left| \varsigma _{d}\right| \)) region. Taking into account phenomenological constraints from the inclusive radiative decay \(B\rightarrow X_{s}{\gamma }\), as well as the latest model-independent global analysis of \(b\rightarrow s\ell ^+\ell ^-\) data, we obtain the much restricted parameter space of the model. We then study the impact of the allowed model parameters on the angular observables \(P_2\) and \(P_5'\) of \(B^0\rightarrow K^{*0}\mu ^+\mu ^-\) decay, and we find that \(P_5'\) could be increased significantly to be consistent with the experimental data in case B.     

Near Critical Preferential Attachment Networks have Small Giant Components

Preferential attachment networks with power law exponent \(\tau >3\) are known to exhibit a phase transition. There is a value \(\rho _{\mathrm{c}}>0\) such that, for small edge densities \(\rho \le \rho _{\mathrm{c}}\) every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities \(\rho >\rho _{\mathrm{c}}\) there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like \(\exp (-c/ \sqrt{\rho -\rho _{\mathrm{c}}})\) for an explicit constant \(c>0\) depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).     

Higgs–photon resonances

We study models that produce a Higgs boson plus photon (\(h^0\gamma \)) resonance at the LHC. When the resonance is a \(Z'\) boson, decays to \(h^0\gamma \) occur at one loop. If the \(Z'\) boson couples at tree level to quarks, then the \(h^0\gamma \) branching fraction is typically of order \(10^{-5}\) or smaller. Nevertheless, there are models that would allow the observation of \(Z'\rightarrow \,h^0\gamma \) at \(\sqrt{s}=13\) TeV with a cross section times branching fraction larger than 1 fb for a \(Z'\) mass in the 200–450 GeV range, and larger than 0.1 fb for a mass up to 800 GeV. The one-loop decay of the \(Z'\) into lepton pairs competes with \(h^0\gamma \), even if the \(Z'\) couplings to leptons vanish at tree level. We also present a model in which a \(Z'\) boson decays into a Higgs boson and a pair of collimated photons, mimicking an \(h^0\gamma \) resonance. In this model, the \(h^0\gamma \) resonance search would be the discovery mode for a \(Z'\) as heavy as 2 TeV. When the resonance is a scalar, although decay to \(h^0\gamma \) is forbidden by angular momentum conservation, the \(h^0\) plus collimated photons channel is allowed. We comment on prospects of observing an \(h^0\gamma \) resonance through different Higgs decays, on constraints from related searches, and on models where \(h^0\) is replaced by a nonstandard Higgs boson.     

Graph-Counting Polynomials for Oriented Graphs

If \(\mathcal{F}\) is a set of subgraphs F of a finite graph E we define a graph-counting polynomial \(p_\mathcal{F}(z)=\sum _{F\in \mathcal{F}}z^{|F|}\) In the present note we consider oriented graphs and discuss some cases where \(\mathcal{F}\) consists of unbranched subgraphs E. We find several situations where something can be said about the location of the zeros of \(p_\mathcal{F}\).     

Warm modified Chaplygin gas shaft inflation

In this paper, we examine the possible realization of a new inflation family called “shaft inflation” by assuming the modified Chaplygin gas model and a tachyon scalar field. We also consider the special form of the dissipative coefficient \(\Gamma ={a_0}\frac{T^{3}}{\phi ^{2 }}\) and calculate the various inflationary parameters in the scenario of strong and weak dissipative regimes. In order to examine the behavior of inflationary parameters, the \(n_s \)–\( \phi ,\, n_s \)–r, and \(n_s \)–\( \alpha _s\) planes (where \(n_s,\, \alpha _s,\, r\), and \(\phi \) represent the spectral index, its running, tensor-to-scalar ratio, and scalar field, respectively) are being developed, which lead to the constraints \(r< 0.11\), \(n_s=0.96 \pm 0.025\), and \(\alpha _s =-0.019 \pm 0.025\). It is quite interesting that these results of the inflationary parameters are compatible with BICEP2, WMAP \((7+9)\) and recent Planck data.     

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional \({\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x\) under the mass constraint \(\int _{{\mathbb {R}}^3}|u(x)|^2\mathrm{d}x=1\). Here \(m>0\) is the mass of particles and \(V\ge 0\) is an external potential. We prove that minimizers exist if and only if a satisfies \(0\le a<a^{*}\), and there is no minimizer if \(a\ge a^*\), where \(a^*\) is called the Chandrasekhar limit. When a approaches \(a^*\) from below, the blow-up behavior of minimizers is derived under some general external potentials V. Here we consider three cases of V: trapping potential, i.e. \(V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)\) satisfies \(\lim _{|x|\rightarrow \infty }V(x)=\infty \); periodic potential, i.e. \(V\in C({\mathbb {R}}^3)\) satisfies \(V(x+z)=V(x)\) for all \(z\in \mathbb {Z}^3\); and ring-shaped potential, e.g. \( V(x)=||x|-1|^p\) for some \(p>0\).     

All static and electrically charged solutions with Einstein base manifold in the arbitrary-dimensional Einstein–Maxwell system with a massless scalar field

We present a simple and complete classification of static solutions in the Einstein–Maxwell system with a massless scalar field in arbitrary \(n(\ge 3)\) dimensions. We consider spacetimes which correspond to a warped product \(M^2 \times K^{n-2}\), where \(K^{n-2}\) is a \((n-2)\)-dimensional Einstein space. The scalar field is assumed to depend only on the radial coordinate and the electromagnetic field is purely electric. Suitable Ansätze enable us to integrate the field equations in a general form and express the solutions in terms of elementary functions. The classification with a non-constant real scalar field consists of nine solutions for \(n\ge 4\) and three solutions for \(n=3\). A complete geometric analysis of the solutions is presented and the global mass and electric charge are determined for asymptotically flat configurations. There are two remarkable features for the solutions with \(n\ge 4\): (i) Unlike the case with a vanishing electromagnetic field or constant scalar field, asymptotically flat solution is not unique, and (ii) The solutions can asymptotically approach the Bertotti–Robinson spacetime depending on the integrations constants. In accordance with the no-hair theorem, none of the solutions are endowed of a Killing horizon.     

1 / <Emphasis Type="Italic">n</Emphasis> Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy

Using a 1 / n expansion, that is an expansion in descending powers of n, for the number of matchings in regular graphs with 2n vertices, we study the monomer-dimer entropy for two classes of graphs. We study the difference between the extensive monomer-dimer entropy of a random r-regular graph G (bipartite or not) with 2n vertices and the average extensive entropy of r-regular graphs with 2n vertices, in the limit \(n \rightarrow \infty \). We find a series expansion for it in the numbers of cycles; with probability 1 it converges for dimer density \(p < 1\) and, for G bipartite, it diverges as \(|\mathrm{ln}(1-p)|\) for \(p \rightarrow 1\). In the case of regular lattices, we similarly expand the difference between the specific monomer-dimer entropy on a lattice and the one on the Bethe lattice; we write down its Taylor expansion in powers of p through the order 10, expressed in terms of the number of totally reducible walks which are not tree-like. We prove through order 6 that its expansion coefficients in powers of p are non-negative.     

Universality for 1d Random Band Matrices: Sigma-Model Approximation

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of \(W\times W\) random Gaussian blocks (parametrized by \(j,k \in \Lambda =[1,n]^d\cap \mathbb {Z}^d\)) with a fixed entry’s variance \(J_{jk}=\delta _{j,k}W^{-1}+\beta \Delta _{j,k}W^{-2}\), \(\beta >0\) in each block. Taking the limit \(W\rightarrow \infty \) with fixed n and \(\beta \), we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit \(\beta , n\rightarrow \infty \), we prove that in the dimension \(d=1\) the behaviour of the sigma-model approximation in the bulk of the spectrum, as \(\beta \gg n\), is determined by the classical Wigner–Dyson statistics.     

When is a Scale-Free Graph Ultra-Small?

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent \(\tau \in (2,3)\), up to value \(n^{{\beta _n}}\) for some \({\beta _n}\gg (\log n)^{-\gamma }\) and \(\gamma \in (0,1)\). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at \(n^{{\beta _n}}\). These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around \(2 \log \log (n^{{\beta _n}}) / |\log (\tau -2)| + 1/({\beta _n}(3-\tau ))\), with tight fluctuations. Thus, the graph is an ultrasmall world whenever \(1/{\beta _n}=o(\log \log n)\). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show \(\log \log \)-periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order \(n^{f_n{\beta _n}}\), where \(f_n \in (0,1)\) is a random variable that oscillates with n. We decompose shortest paths into three segments, two ‘end-segments’ starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length \(\log \log (n^{{\beta _n}}) / |\log (\tau -2)|\)+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length \(1/({\beta _n}(3-\tau ))\)+tight, and it contains only vertices with degree at least of order \(n^{(1-f_n){\beta _n}}\), thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least \(n^{{\beta _n}}\), and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.