Weak and Strong Connectivity Regimes for a General Time Elapsed Neuron Network Model

For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works (Pakdaman et al. in Nonlinearity 23(1):55–75, 2010; SIAM J Appl Math 73(3):1260–1279, 2013, Mischler and Weng in Acta Appl Math, 2015), we establish accurate estimate on the long time behavior of the solutions in the weak and the strong connectivity regime both in the case with and without delay. Our results improve (Pakdaman et al. 2010, 2013) where a less accurate estimate was established and Mischler and Weng (2015) where only smooth firing rates were considered. Our approach combines several arguments introduced in the above previous works as well as a slightly refined version of the Weyl’s and spectral mapping theorems presented in Voigt (Monatsh Math 90(2):153–161, 1980) and Mischler and Scher (Ann Inst H Poincaré Anal Non Linéaire 33(3):849–898, 2016).     

Reconstruction of a kinetic k-essence Lagrangian from a modified of dark energy equation of state

In this paper the Lagrangian density of a purely kinetic k-essence model that describes the behavior of dark energy described by four parameterized equations of state proposed by Cooray and Huterer (Astrophys J 513:L95, 1999), Zhang and Wu (Mod Phys Lett A 27:1250030, 2012), Linder (Phys Rev Lett 90:091301, 2003), Efstathiou (Mon Not R Astron Soc 310:842, 2000), and Feng and Lu (J Cosmol Astropart Phys 1111:34, 2011) has been reconstructed. This reconstruction is performed using the method outlined by de Putter and Linder (Astropart Phys 28:263, 2007), which makes it possible to solve the equations that relate the Lagrangian density of the k-essence with the given equation of state (EoS) numerically. Finally, we discuss the observational constraints for the models based on 1049 SNIa data points from the Pantheon data set compiled by Scolnic et al. (Astrophys J 859(2):101, 2018)     

Gravitational Rutherford scattering and Keplerian orbits for electrically charged bodies in heterotic string theory

Properties of the motion of electrically charged particles in the background of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole is presented in this paper. Radial and angular motions are studied analytically for different values of the fundamental parameter. Therefore, gravitational Rutherford scattering and Keplerian orbits are analyzed in detail. Finally, this paper complements previous work by Fernando for null geodesics (Phys Rev D 85:024033, 2012), Olivares and Villanueva (Eur Phys J C 73:2659, 2013) and Blaga (Automat Comp Appl Math 22:41–48, 2013; Serb Astron 190:41, 2015) for time-like geodesics.     

A Pfaffian formula for the monomer–dimer model on surface graphs

We consider the monomer–dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani et al. (J Stat Phys 163(2):211–238, 2016) for graphs embedded in the disc. Our proof is based on an extension of a bijective method mentioned in Giuliani et al. (2016), together with the Pfaffian formula for the dimer partition function of Cimasoni–Reshetikhin (Commun Math Phys 275(1):187–208, 2007).     

A Second Order Expansion of the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems

In this paper we study a so-called separatrix map introduced by Zaslavskii–Filonenko (Sov Phys JETP 27:851–857, 1968) and studied by Treschev (Physica D 116(1–2):21–43, 1998; J Nonlinear Sci 12(1):27–58, 2002), Piftankin (Nonlinearity (19):2617–2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3–108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at http://www.terpconnect.umd.edu/vkaloshi/, 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.     

Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems

In two papers Franz et al. proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution (Franz and Leone in J Stat Phys 111(3–4):535–564, 2003) and a general distribution (Franz et al. in J Phys A 36(43), 10967, 2003). Panchenko and Talagrand (Probab Theo Relat Fields 130(3):319–336, 2004) simplified the proof and generalized the result of Franz and Leone (J Stat Phys 111(3–4):535–564, 2003) for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks (Salez in Combin Probab Comput 25(03):436–447, 2016) and allows to bypass the arguments in Franz et al. (J Phys A 36(43):10967, 2003) based on the study of the Sherrington–Kirkpatrick (SK) model.     

The Corolla Polynomial for Spontaneously Broken Gauge Theories

In Kreimer and Yeats (Electr. J. Comb. 41–41, 2013), Kreimer et al. (Annals Phys. 336, 180–222, 2013) and Sars (2015) the Corolla Polynomial \( \mathcal C ({\Gamma }) \in \mathbb C [a_{h_{1}}, \ldots , a_{h_{\left \vert {\Gamma }^{[1/2]} \right \vert }}]\) was introduced as a graph polynomial in half-edge variables \(\{a_{h}\}_{h \in {\Gamma }^{[1/2]}}\) over a 3-regular scalar quantum field theory (QFT) Feynman graph Γ. It allows for a covariant quantization of pure Yang-Mills theory without the need for introducing ghost fields, clarifies the relation between quantum gauge theory and scalar QFT with cubic interaction and translates back the problem of renormalizing quantum gauge theory to the problem of renormalizing scalar QFT with cubic interaction (which is super renormalizable in 4 dimensions of spacetime). Furthermore, it is, as we believe, useful for computer calculations. In Prinz (2015) on which this paper is based the formulation of Kreimer and Yeats (Electr. J. Comb. 41–41, 2013), Kreimer et al. (Annals Phys. 336, 180–222, 2013) and Sars (2015) gets slightly altered in a fashion specialized in the case of the Feynman gauge. It is then formulated as a graph polynomial \(\mathcal C ({\Gamma } ) \in \mathbb C [a_{h_{1 \pm }}, \ldots , a_{h_{\left \vert {\Gamma }^{[1/2]} \right \vert } \vphantom {h}_{\pm }}, b_{h_{1}}, \ldots , b_{h_{\left \vert {\Gamma }^{[1/2]} \right \vert }}] \) in three different types of half-edge variables \( \{a_{h_{+}} , a_{h_{-}} , b_{h}\}_{h \in {\Gamma }^{[1/2]}} \). This formulation is also suitable for the generalization to the case of spontaneously broken gauge theories (in particular all bosons from the Standard Model), as was first worked out in Prinz (2015) and gets reviewed here.     

Convergence of Mayer and Virial expansions and the Penrose tree-graph identity

We establish new lower bounds for the convergence radius of the Mayer series and the Virial series of a continuous particle system interacting via a stable and tempered pair potential. Our bounds considerably improve those given by Penrose (J Math Phys 4:1312, 1963) and Ruelle (Ann Phys 5:109–120, 1963) for the Mayer series and by Lebowitz and Penrose (J Math Phys 7:841–847, 1964) for the Virial series. To get our results, we exploit the tree-graph identity given by Penrose (Statistical mechanics: foundations and applications. Benjamin, New York, 1967) using a new partition scheme based on minimum spanning trees.     

Two-Dimensional Anisotropic Random Walks: Fixed Versus Random Column Configurations for Transport Phenomena

We consider random walks on the square lattice of the plane along the lines of Heyde (J Stat Phys 27:721–730, 1982, Stochastic processes, Springer, New York, 1993) and den Hollander (J Stat Phys 75:891–918, 1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions á  la Heyde (J Stat Phys 27:721–730, 1982, Stochastic processes, Springer, New York, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (J Stat Phys 75:891–918, 1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context.     

Contiguity and Entire Separability of States on von Neumann Algebras

We introduce the notions of the contiguity and entirely separability for two sequences of states on von Neumann algebras. The ultraproducts technique allows us to reduce the study of the contiguity to investigation of the equivalence for two states. Here we apply the Ocneanu ultraproduct and the Groh–Raynaud ultraproduct (see Ocneanu (1985), Groh (J. Operator Theory, 11, 2, 395–404 1984), Raynaud (J. Operator Theory, 48, 1, 41–68, 2002), Ando and Haagerup (J. Funct. Anal., 266, 12, 6842–6913, 2014)), as well as the technique developed in Mushtari and Haliullin (Lobachevskii J. Math., 35, 2, 138–146, 2014).     

Finite vs. Infinite Decompositions in Conformal Embeddings

We revisit two old and apparently little known papers by Basuev (Teoret Mat Fiz 37(1):130–134, 1978, Teoret Mat Fiz 39(1):94–105, 1979) and show that the results contained there yield strong improvements on current lower bounds of the convergence radius of the Mayer series for continuous particle systems interacting via a very large class of stable and tempered potentials, which includes the Lennard-Jones type potentials. In particular we analyze the case of the classical Lennard-Jones gas under the light of the Basuev scheme and, using also some new results (Yuhjtman in J Stat Phys 160(6): 1684–1695, 2015) on this model recently obtained by one of us, we provide a new lower bound for the Mayer series convergence radius of the classical Lennard-Jones gas, which improves by a factor of the order 10⁵ on the current best lower bound recently obtained in de Lima and Procacci (J Stat Phys 157(3):422–435, 2014).     

Totally Asymmetric Limit for Models of Heat Conduction

We consider one dimensional weakly asymmetric boundary driven models of heat conduction. In the cases of a constant diffusion coefficient and of a quadratic mobility we compute the quasi-potential that is a non local functional obtained by the solution of a variational problem. This is done using the dynamic variational approach of the macroscopic fluctuation theory (Bertini et al. in Rev Mod Phys 87:593, 2015). The case of a concave mobility corresponds essentially to the exclusion model that has been discussed in Bertini et al. (J Stat Mech L11001, 2010; Pure Appl Math 64(5):649–696, 2011; Commun Math Phys 289(1):311–334, 2009) and Enaud and Derrida (J Stat Phys 114:537–562, 2004). We consider here the convex case that includes for example the Kipnis-Marchioro-Presutti (KMP) model and its dual (KMPd) (Kipnis et al. in J Stat Phys 27:6574, 1982). This extends to the weakly asymmetric regime the computations in Bertini et al. (J Stat Phys 121(5/6):843–885, 2005). We consider then, both microscopically and macroscopically, the limit of large externalfields. Microscopically we discuss some possible totally asymmetric limits of the KMP model. In one case the totally asymmetric dynamics has a product invariant measure. Another possible limit dynamics has instead a non trivial invariant measure for which we give a duality representation. Macroscopically we show that the quasi-potentials of KMP and KMPd, which are non local for any value of the external field, become local in the limit. Moreover the dependence on one of the external reservoirs disappears. For models having strictly positive quadratic mobilities we obtain instead in the limit a non local functional having a structure similar to the one of the boundary driven asymmetric exclusion process.     

Constraints on tidal charge of the supermassive black hole at the Galactic Center with trajectories of bright stars

As it was pointed out recently in Hees et al. (Phys Rev Lett 118:211101, 2017), observations of stars near the Galactic Center with current and future facilities provide an unique tool to test general relativity (GR) and alternative theories of gravity in a strong gravitational field regime. In particular, the authors showed that the Yukawa gravity could be constrained with Keck and TMT observations. Some time ago, Dadhich et al. (Phys Lett B 487:1, 2001) showed that the Reissner–Nordström metric with a tidal charge is naturally appeared in the framework of Randall–Sundrum model with an extra dimension (\(Q^2\) is called tidal charge and it could be negative in such an approach). Astrophysical consequences of presence of black holes with a tidal charge are considerered, in particular, geodesics and shadows in Kerr–Newman braneworld metric are analyzed in Schee and Stuchlík (Intern J Mod Phys D 18:983, 2009), while profiles of emission lines generated by rings orbiting braneworld Kerr black hole are considered in Schee and Stuchlík (Gen Relat Grav 52:1795, 2009). Possible observational signatures of gravitational lensing in a presence of the Reissner–Nordström black hole with a tidal charge at the Galactic Center are discussed in papers (Bin-Nun in Phys Rev D 81:123011, 2010; Bin-Nun in Phys Rev D 82:064009, 2010; Bin-Nun in Class Quant Grav 28:114003, 2011). Here we are following such an approach and we obtain analytical expressions for orbital precession for Reissner–Nordström–de-Sitter solution in post-Newtonian approximation and discuss opportunities to constrain parameters of the metric from observations of bright stars with current and future astrometric observational facilities such as VLT, Keck, GRAVITY, E-ELT and TMT.     

Yes,More Decoherence: A Reply to Critics

Recently I published an article in this journal entitled “Less interpretation and more decoherence in quantum gravity and inflationary cosmology” (Crull in Found Phys 45(9):1019–1045, 2015). This article generated responses from three pairs of authors: Vassallo and Esfeld (Found Phys 45(12):1533–1536, 2015), Okon and Sudarsky (Found Phys 46(7):852–879, 2016) and Fortin and Lombardi (Found Phys, 2017). In what follows, I reply to the criticisms raised by these authors.     

The Quest for the Ultimate Anisotropic Banach Space

We present a new scale \(\mathcal {U}^{t,s}_p\) (\(s<-t<0\) and \(1\le p <\infty \)) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator \(\mathcal {L}_g \varphi = (g \cdot \varphi ) \circ T^{-1}\) associated to a hyperbolic dynamical system T has good spectral properties. When \(p=1\) and t is an integer, the spaces are analogous to the “geometric” spaces \(\mathcal {B}^{t,|s+t|}\) considered by Gouëzel and Liverani (Ergod Theory Dyn Syst 26:189–217, 2006). When \(p>1\) and \(-1+1/p<s<-t<0<t<1/p\), the spaces are somewhat analogous to the geometric spaces considered by Demers and Liverani (Trans Am Math Soc 360:4777–4814, 2008). In addition, just like for the “microlocal” spaces defined by Baladi and Tsujii (Ann Inst Fourier 57:127–154, 2007) (or Faure–Roy–Sjöstrand in Open Math J 1:35–81, 2008), the transfer operator acting on \(\mathcal {U}^{t,s}_p\) can be decomposed into \(\mathcal {L}_{g,b}+\mathcal {L}_{g,c}\), where \(\mathcal {L}_{g,b}\) has a controlled norm while a suitable power of \(\mathcal {L}_{g,c}\) is nuclear. This “nuclear power decomposition” enhances the Lasota–Yorke bounds and makes the spaces \(\mathcal {U}^{t,s}_p\) amenable to the kneading approach of Milnor–Thurson (Dynamical Systems (Maryland 1986–1987), Springer, Berlin, 1988) (as revisited by Baladi–Ruelle, Baladi in Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Monograph, 2016; Baladi and Ruelle in Ergod Theory Dyn Syst 14:621–632, 1994; Baladi and Ruelle in Invent Math 123:553–574, 1996) to study dynamical determinants and zeta functions.     

Comment on “Concrete Representation and Separability Criteria for Symmetric Quantum State”

Recently, Li et al. (Int. J. Theor. Phys. 53(9), 2923–2930 (2014)) presented the concrete representation of density matrix of symmetric quantum states . Moreover , according to this concrete representation of the density matrix for symmetric quantum states, Li et al. (Int. J. Theor. Phys. 53(9), 2923–2930 (2014)) have established Theorem 4.1. In this Comment, we would like to point out that Theorem 4.1 given by Li et al. (Int. J. Theor. Phys. 53(9), 2923–2930 (2014)) is incorrect in general.     

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

We study the persistence probability for some two-sided, discrete-time Gaussian sequences that are discrete-time analogues of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous time in Molchan (Commun Math Phys 205(1):97–111, 1999) and Molchan (J Stat Phys 167(6):1546–1554, 2017) to a wide class of discrete-time processes.     

Load Balancing in Hypergraphs

Consider a simple locally finite hypergraph on a countable vertex set, where each edge represents one unit of load which should be distributed among the vertices defining the edge. An allocation of load is called balanced if load cannot be moved from a vertex to another that is carrying less load. We analyze the properties of balanced allocations of load. We extend the concept of balancedness from finite hypergraphs to their local weak limits in the sense of Benjamini and Schramm (Electron J Probab 6(23):13, 2001) and Aldous and Steele (in: Probability on discrete structures. Springer, Berlin, pp 1–72, 2004). To do this, we define a notion of unimodularity for hypergraphs which could be considered an extension of unimodularity in graphs. We give a variational formula for the balanced load distribution and, in particular, we characterize it in the special case of unimodular hypergraph Galton–Watson processes. Moreover, we prove the convergence of the maximum load under some conditions. Our work is an extension to hypergraphs of Anantharam and Salez (Ann Appl Probab 26(1):305–327, 2016), which considered load balancing in graphs, and is aimed at more comprehensively resolving conjectures of Hajek (IEEE Trans Inf Theory 36(6):1398–1414, 1990).     

Random Forests and Networks Analysis

Wilson (Proceedings of the twenty-eight annual acm symposium on the theory of computing, pp. 296–303, 1996) in the 1990s described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works (Avena and Gaudillière in A proof of the transfer-current theorem in absence of reversibility, in Stat. Probab. Lett. 142, 17–22 (2018); Avena and Gaudillière in J Theor Probab, 2017.  https://doi.org/10.1007/s10959-017-0771-3; Avena et al. in Approximate and exact solutions of intertwining equations though random spanning forests, 2017. arXiv:1702.05992v1; Avena et al. in Intertwining wavelets or multiresolution analysis on graphs through random forests, 2017. arXiv:1707.04616, to appear in ACHA (2018)) we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: (1) a random-walk-based notion of well-distributed points in a graph, (2) a framework to describe metastable-like dynamics in finite settings by means of Markov intertwining dualities, (3) coarse graining schemes for networks and associated processes, (4) wavelets-like pyramidal algorithms for graph signals.     

Traces of Intertwiners for Quantum Affine {\mathfrak{sl}}_2 and Felder–Varchenko Functions

We show that the traces of \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-intertwiners of [ESV02] valued in the three-dimensional evaluation representation converge in a certain region of parameters and give a representation-theoretic construction of Felder–Varchenko’s hypergeometric solutions to the q-KZB heat equation given in [FV02]. This gives the first proof that such a trace function converges and resolves the first case of the Etingof–Varchenko conjecture of [EV00]. As applications, we prove a symmetry property for traces of intertwiners and prove Felder–Varchenko’s conjecture in [FV04] that their elliptic Macdonald polynomials are related to the affine Macdonald polynomials defined as traces over irreducible integrable \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-modules in [EK95]. In the trigonometric and classical limits, we recover results of [EK94,EV00]. Our method relies on an interplay between the method of coherent states applied to the free field realization of the q-Wakimoto module of [Mat94], convergence properties given by the theta hypergeometric integrals of [FV02], and rationality properties originating from the representation-theoretic definition of the trace function.