Random Repeated Interaction Quantum Systems

We consider a quantum system \({\mathcal{S}}\) interacting sequentially with independent systems \({\mathcal{E}_m}\) , m = 1,2,... Before interacting, each \({\mathcal{E}_m}\) is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics. We solve exactly a model in which \({\mathcal{S}}\) and all the \({\mathcal{E}_m}\) are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the \({\mathcal{E}_m}\) vary randomly. We analyze a model in which \({\mathcal{S}}\) is a spin and the \({\mathcal{E}_m}\) are thermal fermion baths and obtain the asymptotic state by rigorous perturbation theory, for random interaction times varying slightly around a fixed mean, and for small values of a coupling constant.     

The Vlasov–Poisson System for Stellar Dynamics in Spaces of Constant Curvature

We obtain a natural extension of the Vlasov–Poisson system for stellar dynamics to spaces of constant Gaussian curvature \({\kappa \ne 0}\): the unit sphere \({\mathbb S^2}\), for \({\kappa > 0}\), and the unit hyperbolic sphere \({\mathbb H^2}\), for \({\kappa < 0}\). These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov–Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping.     

Comparison of the DeWitt metric in general relativity with the fourth-rank constitutive tensors in electrodynamics and in elasticity theory

We perform a short comparison between the local and linear constitutive tensor \(\chi ^{\lambda \nu \sigma \kappa }\) in four-dimensional electrodynamics, the elasticity tensor \(c^{ijkl}\) in three-dimensional elasticity theory, and the DeWitt metric \(G^{abcd}\) in general relativity, with \({a,b,\ldots =1,2,3}\). We find that the DeWitt metric has only six independent components.     

Search of the neutrino-less double beta decay of $$^{}$$Se into the excited states of $$^{}$$Kr with CUPID-0

The CUPID-0 experiment searches for double beta decay using cryogenic calorimeters with double (heat and light) read-out. The detector, consisting of 24 ZnSe crystals 95\(\%\) enriched in \(^{82}\)Se and two natural ZnSe crystals, started data-taking in 2017 at Laboratori Nazionali del Gran Sasso. We present the search for the neutrino-less double beta decay of \(^{82}\)Se into the 0\(_1^+\), 2\(_1^+\) and 2\(_2^+\) excited states of \(^{82}\)Kr with an exposure of 5.74 kg\(\cdot \)yr (2.24\(\times \)10\(^{25}\) emitters\(\cdot \)yr). We found no evidence of the decays and set the most stringent limits on the widths of these processes: \(\varGamma \)(\(^{82}\)Se \(\rightarrow ^{82}\)Kr\(_{0_1^+}\))8.55\(\times \)10\(^{-24}\) yr\(^{-1}\), \(\varGamma \) (\(^{82}\) Se \(\rightarrow ^{82}\) Kr \(_{2_1^+}\))\(\,{<}\,6.25 \,{\times }\,10^{-24}\) yr\(^{-1}\), \(\varGamma \)(\(^{82}\)Se \(\rightarrow ^{82}\)Kr\(_{2_2^+}\))8.25\(\times \)10\(^{-24}\) yr\(^{-1}\) (90\(\%\) credible interval).     

Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild–de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle \(\alpha \) of the light ray by constructing a quadrilateral \(\varSigma ^4\) on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) determined by the optical metric \(\bar{g}_{ij}\). On the basis of the definition of the total deflection angle \(\alpha \) and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle \(\alpha \); (1) the angular formula that uses four angles determined on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) or the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\) being a slice of constant time t and (2) the integral formula on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) which is the areal integral of the Gaussian curvature K in the area of a quadrilateral \(\varSigma ^4\) and the line integral of the geodesic curvature \(\kappa _g\) along the curve \(C_{\varGamma }\). As the curve \(C_{\varGamma }\), we introduce the unperturbed reference line that is the null geodesic \(\varGamma \) on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting \(\varGamma \) vertically onto the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\). We demonstrate that the two formulas give the same total deflection angle \(\alpha \) for the Schwarzschild and the Schwarzschild–de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein–Shapiro’s formula when the source S and the receiver R of the light ray are located at infinity. In addition, in the Schwarzschild–de Sitter case, there appear order \({\mathscr {O}}(\varLambda m)\) terms in addition to the Schwarzschild-like part, while order \({\mathscr {O}}(\varLambda )\) terms disappear.     

Discrete One-Dimensional Coverage Process on a Renewal Process

Consider the following coverage model on \(\mathbb {N}\), for each site \(i \in \mathbb {N}\) associate a pair \((\xi _i, R_i)\) where \((\xi _i)_{i \ge 0}\) is a 1-dimensional undelayed discrete renewal point process and \((R_i)_{i \ge 0}\) is an i.i.d. sequence of \(\mathbb {N}\)-valued random variables. At each site where \(\xi _i=1\) start an interval of length \(R_i\). Coverage occurs if every site of \(\mathbb {N}\) is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.     

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.     

Tensor functors between Morita duals of fusion categories

Given a fusion category \({\mathcal C}\) and an indecomposable \({\mathcal C}\)-module category \({\mathcal M}\), the fusion category \({\mathcal C}^*_{_{{\mathcal M}}}\) of \({\mathcal C}\)-module endofunctors of \({\mathcal M}\) is called the (Morita) dual fusion category of \({\mathcal C}\) with respect to \({\mathcal M}\). We describe tensor functors between two arbitrary duals \({\mathcal C}^*_{_{{\mathcal M}}}\) and \({\mathcal D}^*_{\mathcal N}\) in terms of data associated to \({\mathcal C}\) and \({\mathcal D}\). We apply the results to G-equivariantizations of fusion categories and group-theoretical fusion categories. We describe the orbits of the action of the Brauer–Picard group on the set of module categories and we propose a categorification of the Rosenberg–Zelinsky sequence for fusion categories.     

Clustering in the Three and Four Color Cyclic Particle Systems in One Dimension

We study the \(\kappa \)-color cyclic particle system on the one-dimensional integer lattice \(\mathbb {Z}\), first introduced by Bramson and Griffeath (Ann Prob:26–45, 1989). In that paper they show that almost surely, every site changes its color infinitely often if \(\kappa \in \{3,4\}\) and only finitely many times if \(\kappa \ge 5\). In addition, they conjecture that for \(\kappa \in \{3,4\}\) the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as \(t\rightarrow \infty \), x and y have the same color at time t. Here we prove that conjecture.     

Eigenvalue Order Statistics for Random Schrödinger Operators with Doubly-Exponential Tails

We consider random Schrödinger operators of the form \({\Delta+\xi}\), where \({\Delta}\) is the lattice Laplacian on \({\mathbb{Z}^{d}}\) and \({\xi}\) is an i.i.d. random field, and study the extreme order statistics of the Dirichlet eigenvalues for this operator restricted to large but finite subsets of \({\mathbb{Z}^{d}}\). We show that, for \({\xi}\) with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where \({\xi}\) takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Notwithstanding, our approach is largely independent of existing methods for proofs of Anderson localization and it is based on studying individual eigenvalue/eigenfunction pairs and characterizing the regions where the leading eigenfunctions put most of their mass.     

Scattering of kinks of the sinh-deformed $$\varphi ^4$$ model

We consider the scattering of kinks of the sinh-deformed \(\varphi ^4\) model, which is obtained from the well-known \(\varphi ^4\) model by means of the deformation procedure. Depending on the initial velocity \(v_\mathrm {in}\) of the colliding kinks, different collision scenarios are realized. There is a critical value \(v_\mathrm {cr}\) of the initial velocity, which separates the regime of reflection (at \(v_\mathrm {in}>v_\mathrm {cr}\)) and that of a complicated interaction (at \(v_\mathrm {in}<v_\mathrm {cr}\)) with kinks’ capture and escape windows. Besides that, at \(v_\mathrm {in}\) below \(v_\mathrm {cr}\) we observe the formation of a bound state of two oscillons, as well as their escape at some values of \(v_\mathrm {in}\).     

A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

In earlier papers, we have studied the turbulent flow exponents \(\zeta _p\), where \(\langle |\Delta \mathbf{v}|^p\rangle \sim \ell ^{\zeta _p}\) and \(\Delta \mathbf{v}\) is the contribution to the fluid velocity at small scale \(\ell \). Using ideas of non-equilibrium statistical mechanics we have found

$$\begin{aligned} \zeta _p={p\over 3}-{1\over \ln \kappa }\ln \Gamma \left( {p\over 3}+1\right) \end{aligned}$$

where \(1/\ln \kappa \) is experimentally \(\approx \,0.32\,\pm \,0.01\). The purpose of the present note is to propose a somewhat more physical derivation of the formula for \(\zeta _p\). We also present an estimate \(\approx \,100\) for the Reynolds number at the onset of turbulence.

     

Decay widths of the spin-2 partners of the <Emphasis Type="Italic">X</Emphasis>(3872)

We consider the X(3872) resonance as a \(J^\mathrm{{PC}}=1^{++}\) \(D\bar{D}^*\) hadronic molecule. According to heavy quark spin symmetry, there will exist a partner with quantum numbers \(2^{++}\), \(X_{2}\), which would be a \(D^*\bar{D}^*\) loosely bound state. The \(X_{2}\) is expected to decay dominantly into \(D\bar{D}\), \(D\bar{D}^*\) and \(\bar{D} D^*\) in d-wave. In this work, we calculate the decay widths of the \(X_{2}\) resonance into the above channels, as well as those of its bottom partner, \(X_{b2}\), the mass of which comes from assuming heavy flavor symmetry for the contact terms. We find partial widths of the \(X_{2}\) and \(X_{b2}\) of the order of a few MeV. Finally, we also study the radiative \(X_2\rightarrow D\bar{D}^{*}\gamma \) and \(X_{b2} \rightarrow \bar{B} B^{*}\gamma \) decays. These decay modes are more sensitive to the long-distance structure of the resonances and to the \(D\bar{D}^{*}\) or \(B\bar{B}^{*}\) final state interaction.     

Fluctuation of Density of States for 1d Schrödinger Operators

We consider the 1d Schrödinger operator with random decaying potential and compute the 2nd term asymptotics of the density of states, which shows substantial differences between the cases \(\alpha > \frac{1}{2}\), \(\alpha < \frac{1}{2}\) and \(\alpha = \frac{1}{2}\).     

Generic Rigidity for Circle Diffeomorphisms with Breaks

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).     

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.     

On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I

We study the determinant \({\det(I-\gamma K_s), 0 < \gamma < 1}\) , of the integrable Fredholm operator K _s acting on the interval (?1, 1) with kernel \({K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}\) . This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature \({\beta=2}\) , in the presence of an external potential \({v=-\frac{1}{2}\ln(1-\gamma)}\) supported on an interval of length \({\frac{2s}{\pi}}\) . We evaluate, in particular, the double scaling limit of \({\det(I-\gamma K_s)}\) as \({s\rightarrow\infty}\) and \({\gamma\uparrow 1}\) , in the region \({0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}\) , for any fixed \({0 < \delta < 1}\) . This problem was first considered by Dyson (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131–146, 1995).     

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.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).