Mathematical Physics, Analysis and Geometry最新文献

We carry out a Kovalevskaya analysis of the Riccati hierarchy. We determine all indicial loci and Kovalevskaya exponents and identify a rigid recursive structure governing how free parameters enter Laurent solutions. We further identify a nontrivial quasi-homogeneous vector field commuting with the hierarchy and use it to obtain an explicit parametrization of all solutions in terms of a single polynomial. Notably, in the two-dimensional case, the same general solution is recovered by the blow-up resolution. Within this parametrization, collisions of poles correspond to degeneration limits of the principal Laurent family, through which lower indicial loci appear. Finally, negative Kovalevskaya exponents are interpreted analytically through annular Laurent expansions, which describe how different collections of poles dominate in different complex regions.

We consider stochastic lattice gases with stationary product weights and a polynomial perturbation vanishing with the system size that leads to condensation. If the density of particles exceeds a critical value the system phase separates into a bulk with homogeneous distribution of particles and a condensed phase. Depending on parameter values, the latter consists of a single macroscopic cluster or a diverging number of independent clusters on a smaller scale. We establish the condensation transition via the equivalence of ensembles and the main novelty is a derivation of the cluster size distribution using size-biased sampling, generalizing previous work on zero-range and inclusion processes. Simulations of zero-range processes illustrate our theoretical results on the condensate scale and size distribution.

The q-Garnier system was first proposed by Sakai, and alternative directions for its discrete time evolutions were provided by Nagao and Yamada. Recently, it was shown that all of those evolution directions are derived from a birational representation of an extended affine Weyl group which arises from the cluster algebraic construction established by Masuda, Okubo and Tsuda. In this article, we investigate a degeneration structure of the fourth order q-Garnier system by using confluences in quivers.

We study the analytic properties of solutions to the q-Painlevé third equation. The q-Painlevé third equation can be represented as a connection preserving deformation of a linear q-difference equation. We investigate the connection problem for this linear q-difference equation and show that the connection matrix is expressed as a product of connection matrices arising from confluent q-hypergeometric equations.

In this paper, we study the critical points of F-Yang-Mills functional on (mathbb {C}P^n), which are called F-Yang-Mills connections, which is a generalization of Yang-Mills connections. We prove that if ((2+frac{2}{n})F''(x)x+(n+1)F'(x)<0), then the weakly stable F-Yang-Mills connection on (mathbb {C}P^n) must be flat. Moreover, if ((2+frac{2}{n})F''(x)x+(n+1)F'(x)=0), we obtain the structure of curvatures corresponding to weakly stable connections. We also show a gap theorem for F-Yang-Mills connections on (mathbb {C}P^n). Our approach is inspired by Bourguignon and Lawson’s study of Yang-Mills stability on spheres.

We study a mixed quantum Ising–XY model on the semi-infinite rooted Cayley tree of order two. For every vertex u, the edge (langle u,(u,1)rangle ) carries an XY interaction and the edge (langle u,(u,2)rangle ) carries an Ising interaction. Using the compatibility criterion for tree-indexed quantum Markov chains and consistently working with the normalized trace, we derive the translation-invariant boundary equation and compute explicitly the associated local transfer operator, namely the one-step partial-trace map which propagates successor boundary data to the parent vertex. We prove that the boundary equation has a unique positive translation-invariant solution for all (J_I,J_{XY}in mathbb {R}) and (beta >0). Hence the model admits a unique translation-invariant quantum Markov chain generated by a positive translation-invariant boundary condition. We also show that the reduced boundary-law dynamics, i.e. the induced finite-dimensional recursion for the boundary-law parameters, has no admissible periodic points of period greater than one and compute the local two-site entanglement on the natural three-site cluster of the tree.

In this work, we develop a root system construction for Cartan centralisers of classical complex semisimple Lie algebras of types B, C and D, and we analyse the associated polynomial Poisson algebras completely and explicitly in the rank-three cases (B_3), (C_3), and (D_3). This rank 3 case is already highly non-trivial. For (B_n), (C_n) and (D_n), we use this new method to present the main division of cases and how the indecomposable polynomials organize into different types that can be understood via the root systems and their properties. An explicit description of all indecomposable polynomials of type (B_n) is provided only for the rank 3. We provide insight how the type A construction changes once the root system contains signed long roots and, in types (B_n) and (C_n), short or doubled roots. Thus, the directed-cycle construction for type (A_{n-1}) becomes a sign-indexed factor construction with permutation-root, long-root, and short-root generators. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. The resulting finite rank-three sets of generators are used to describe the inclusion relations between the corresponding finitely generated Poisson (mathbb C[mathfrak {h}^*])-algebras and to construct algebraic superintegrable systems attached to the reduction chains (mathfrak {h} subset mathfrak {so}(6,mathbb {C})), (mathfrak {h} subset mathfrak {so}(7,mathbb {C})), and (mathfrak {h} subset mathfrak {sp}(6,mathbb {C})).

We consider annihilating random walks on the finite one-dimensional integer torus with deposition of pairs of particles, conditioned on an atypical number of particle jumps. In the thermodynamic limit the process undergoes a dynamical phase transition in the universality class of the one-dimensional quantum Ising model in a transverse field. All cumulants of the time-integrated particle current and of the time-integrated activity, defined as the total number of particle jumps up to some time t, are obtained in closed form to leading order in system size L at the critical point. The generating functions of the cumulants near the critical point are proved to be given by two universal quantities, viz., by the central charge (c=1/2) of the Virasoro algebra that characterizes the Ising universality class and universal scaling functions that are obtained in explicit form.

In this paper, we establish the Boltzmann–Gibbs principle in the (L^p) sense by applying the Littlewood–Paley–Stein inequality. Our model is an asymmetric Ginzburg–Landau interface model on a one-dimensional periodic lattice. Assuming convexity of the potential, we derive detailed error estimates, particularly their dependence on the size of the system and the size of the region on which the sample average is taken. Notably, the estimates are uniform in the strength of the asymmetry.