Abstract: The p-adic diffusion equation is a pseudo differential equation that is formally analogous to the real diffusion equation. The fundamental solutions to pseudo differential equations that generalize the p-adic diffusion equation give rise to p-adic Brownian motions. We show that these stochastic processes are similar to real Brownian motion in that they arise as limits of discrete time random walks on grids. While similar to those in the real case, the random walks in the p-adic setting are necessarily non-local. The study of discrete time random walks that converge to Brownian motion provides intuition about Brownian motion that is important in applications and such intuition is now available in a non-Archimedean setting. PubDate: 2019-07-01

Abstract: Necessary and sufficient conditions are provided for a class of warped product manifolds with non-vanishing flux to be supersymmetric solutions of 11D supergravity. Many non-compact, but complete solutions can be obtained in this manner, including the multi-membrane solution initially found by Duff and Stelle. In a different direction, an explicit 5-parameter moduli space of solutions to 11D supergravity is also constructed which can be viewed as non-supersymmetric deformations of the Duff–Stelle solution. PubDate: 2019-07-01

Abstract: Kinetically constrained models (KCM) are reversible interacting particle systems on \({\mathbb{Z}^{d}}\) with continuous timeMarkov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as \({\mathcal{U}}\) -bootstrap percolation. KCM also display some of the peculiar features of the so-called “glassy dynamics”, and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule \({\mathcal{U}}\) , and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of \({\mathcal{U}}\) -bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of \({\mathcal{U}}\) -bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in Marêché et al. (Exact asymptotics for Duarte and supercritical rooted kinetically constrained models). In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding \({\mathcal{U}}\) -bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics. PubDate: 2019-07-01

Abstract: We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length L, density \(\rho \) , dimension d and jump density \(\varphi \) , one samples \(\rho L^d\) particles in a d-dimensional torus of side length L, and a permutation \(\pi \) of the particles, with probability density proportional to the product of values of \(\varphi \) at the differences between a particle and its image under \(\pi \) . The distribution may be further weighted by a factor of \(\theta \) to the number of cycles in \(\pi \) . Following Matsubara and Feynman, the emergence of macroscopic cycles in \(\pi \) at high density \(\rho \) has been related to the phenomenon of Bose–Einstein condensation. For each dimension \(d\ge 1\) , we identify sub-critical, critical and super-critical regimes for \(\rho \) and find the limiting distribution of cycle lengths in these regimes. The results extend the work of Betz and Ueltschi. Our main technical tools are saddle-point and singularity analysis of suitable generating functions following the analysis by Bogachev and Zeindler of a related surrogate-spatial model. PubDate: 2019-07-01

Abstract: It was known through the efforts of many works that the generating functions in the closed Gromov–Witten theory of \(K_{{\mathbb {P}}^2}\) are meromorphic quasi-modular forms (Coates and Iritani in Kyoto J Math 58(4):695–864, 2018; Lho and Pandharipande in Adv Math 332:349–402, 2018; Coates and Iritani in Gromov–Witten invariants of local \({\mathbb {P}}^{2}\) and modular forms, arXiv:1804.03292 [math.AG], 2018) basing on the B-model predictions (Bershadsky et al. in Commun Math Phys 165:311–428, 1994; Aganagic et al. in Commun Math Phys 277:771–819, 2008; Alim et al. in Adv Theor Math Phys 18(2):401–467, 2014). In this article, we extend the modularity phenomenon to \(K_{{{\mathbb {P}}^1}\times {{\mathbb {P}}^1}}, K_{W{\mathbb {P}}[1,1,2]}, K_{{\mathbb {F}}_1}\) . More importantly, we generalize it to the generating functions in the open Gromov–Witten theory using the theory of Jacobi forms where the open Gromov–Witten parameters are transformed into elliptic variables. PubDate: 2019-07-01

Abstract: Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross–Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross–Pitaevskii equation, but not for any other cases. PubDate: 2019-07-01

Abstract: We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlevé equations (except for PI). In our framework, the Painlevé equations PVI–PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlevé equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four. PubDate: 2019-07-01

Abstract: We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum \({[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]}\) down to the optimal spectral scale, we prove that the Green’s functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten–McKay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. Our method is based on estimating the Green’s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs. PubDate: 2019-07-01

Abstract: We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an \(L_\infty \) algebra. The \(L_\infty \) brackets are given as derived brackets constructed using an underlying Borcherds superalgebra \({\mathscr {B}}({{\mathfrak {g}}}_{r+1})\) , which is a double extension of the structure algebra \({{\mathfrak {g}}}_r\) . The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra. PubDate: 2019-07-01

Abstract: In this paper most of the classes of G2-structures with Einstein induced metric of negative, null, or positive scalar curvature are realized. This is carried out by means of warped G2-structures with fiber an Einstein SU(3) manifold. The torsion forms of any warped G2-structure are explicitly described in terms of the torsion forms of the SU(3)-structure and the warping function, which allows to give characterizations of the principal classes of Einstein warped G2 manifolds. Similar results are obtained for Einstein warped Spin(7) manifolds with fiber a G2 manifold. PubDate: 2019-07-01

Abstract: We consider a particular weak disorder limit (continuum limit) of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst (J Phys A 16:2641–2654, 1983), which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy–Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu (Phys Rev 176:631–643, 1968) and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy–Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is \(C^\infty \) but not analytic at the critical temperature. PubDate: 2019-07-01

Abstract: In the area of quantum chaos, it is of great interest to study the distribution of the \(L^2\) -mass of eigenfunctions of the Laplacian as eigenvalues tend to infinity. Luo and Sarnak first formulated and proved arithmetic quantum unique ergodicity for the continuous spectrum (spanned by Eisenstein series) of the hyperbolic Laplacian on \(SL(2,\mathbb {Z})\backslash \mathbb {H}\) by utilizing the sub-convexity bounds of L-functions associated to Maass cusp forms. In this paper, we build on Luo and Sarnak’s method and explore the structure properties of the constant terms of GL(n) Eisenstein series and extend their results to higher ranks. We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n). PubDate: 2019-07-01

Abstract: Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation. PubDate: 2019-07-01

Abstract: The Lee–Yang property of certain moment generating functions having only pure imaginary zeros is valid for Ising type models with one-component spins and XY models with two-component spins. Villain models and complex Gaussian multiplicative chaos are two-component systems analogous to XY models and related to Gaussian free fields. Although the Lee–Yang property is known to be valid generally in the first case, we show that is not so in the second. Our proof is based on two theorems of general interest relating the Lee–Yang property to distribution tail behavior. PubDate: 2019-07-01

Abstract: The orbifold construction \({A\mapsto A^G}\) for a finite group G is fundamental in rational conformal field theory. The construction of Rep(AG) from Rep(A) on the categorical level, often called gauging, is also prominent in the study of topological phases of matter. Given a non-degenerate braided fusion category \({\mathcal{C}}\) with a G-action, the key step in this construction is to find a braided G-crossed extension compatible with the action. The extension theory of Etingof–Nikshych–Ostrik gives two obstructions for this problem, \({o_3\in H^3(G)}\) and \({o_4\in H^4(G)}\) for certain coefficients, the latter of which depends on a categorical lifting of the action and is notoriously difficult to compute. We show that in the case where \({G\le S_n}\) acts by permutations on \({\mathcal{C}^{\boxtimes n}}\) , both of these obstructions vanish. This verifies a conjecture of Müger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets. PubDate: 2019-07-01

Abstract: In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces. Some other results in this article are convergence of the exploration process of the loop ensemble (or the branch of the exploration tree) to \(\hbox {SLE}(\kappa ,\kappa -6)\) , \(\kappa =16/3\) , and convergence of a generalization of this process for 4 marked points to \(\hbox {SLE}[\kappa ,Z]\) , \(\kappa =16/3\) , where Z refers to a partition function. The latter SLE process is a process that can’t be written as a \(\hbox {SLE}(\kappa ,\rho _1,\rho _2,\ldots )\) process, which are the most commonly considered generalizations of SLEs. PubDate: 2019-07-01

Abstract: The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type ADE carries a structure of P/Q-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of P/Q-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Plücker-type relations in the homogeneous coordinate ring. PubDate: 2019-07-01

Abstract: We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue \(E > 0\) , both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ( \(E \rightarrow \infty\) ). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat 2-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002). PubDate: 2019-07-01

Abstract: Domain wall solitons are basic constructs realizing phase transitions in various field-theoretical models and are solutions to some nonlinear ordinary differential equations descending from the corresponding full sets of governing equations in higher dimensions. In this paper, we present a series of domain wall solitons arising in several classical gauge field theory models. In the context of the Abelian gauge field theory, we unveil the surprising result that the solutions may explicitly be constructed, which enriches our knowledge on integrability of the planar Liouville type equations in their one-dimensional limits. In the context of the non-Abelian gauge field theory, we obtain some existence theorems for domain wall solutions arising in the electroweak type theories by developing some methods of calculus of variations formulated as direct and constrained minimization problems over a weighted Sobolev space. PubDate: 2019-07-01

Abstract: We establish a direct link between Dunkl operators and quantum Lax matrices \({{\mathcal{L}}}\) for the Calogero–Moser systems associated to an arbitrary Weyl group W (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix \({{\mathcal{A}}}\) so that \({{\mathcal{L}}, {\mathcal{A}}}\) form a quantum Lax pair. Moreover, such an \({{\mathcal{A}}}\) can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of W, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D’Hoker–Phong and Bordner–Corrigan–Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic BCn case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem. PubDate: 2019-07-01