Mathematical Physics, Analysis and Geometry最新文献

In this article, we explore the possibility of inheriting an almost Ricci soliton structure on a compact Riemannian manifold (left( M^{n},gright) ) of dimension n through an isometric embedding of (left( M^{n},gright) ) into the Euclidean space (left( R^{m},{overline{g}}right) ), (m>n). For achieving this goal, we choose a constant unit vector (overrightarrow{a}) on (R^{m}) with its tangential component (zeta ) and normal component ({overline{N}}), and call (zeta ) the KN-vector, ({overline{N}}) the KN-normal. We use a lower bound involving a smooth function f on (M^{n}) on the integral of the Ricci curvature (Ricleft( zeta ,zeta right) ) with respect to the KN-vector (zeta ) to show that (left( M^{n},g,zeta ,fright) ) is almost Ricci soliton, which is called the KN-almost Ricci soliton. The mean curvature vector H, gives a natural function (varphi ={overline{g}}left( H,{overline{N}}right) ) on the KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ) called KN-function. Then, we find a condition involving the KN-function (varphi ) to show that an n-dimensional compact proper KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ), (n>2), is isometric to the sphere (S^{n}(c)). In this article, we also find conditions which make a compact KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ) trivial. In first result in this direction, we show that a compact n-dimensional KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ), (n>2), with KN-function (varphi ) and Ricci curvature in the direction of (zeta ) bounded below by (-(n-1)zeta left( varphi right) ) is either isometric to the sphere (S^{n}(c)) or else it is a trivial Ricci soliton. Finally, we show that a compact n-dimensional KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ), (n>2), having scalar curvature (tau ) and KN-function (varphi ) satisfying (tau varphi ge 0) is necessarily a trivial Ricci soliton.

This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate (lambda ), which results in both neighbors attempting to teach each other a fraction (mu ) of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when (lambda le lambda _c) and/or (mu = 0), where (lambda _c) denotes the critical value of the contact process. In contrast, we prove that, for all (lambda > lambda _c), there is a unique phase transition in the direction of (mu ), and for all (mu > 0), there is a unique phase transition in the direction of (lambda ). Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.

Real abstract Wiener spaces (AWS) were originally defined by Gross using measurable norms, as a generalisation of the theory of advanced integral calculus in infinite dimensions as introduced by Cameron and Martin. In this paper we present a rigorous, complete and self-contained general framework for (mathbb {K})-AWS, where (mathbb {K} in {mathbb {R},mathbb {C}}) using the language of characteristic functions instead of measurable norms. In particular, we will prove that X is a symmetric H-valued Gaussian field over (mathbb {K}) iff the covariance function can be written in terms of some non-negative, self-adjoint trace class operator, and that the existence and uniqueness of X is equivalent to the (mathbb {K})-AWS. Finally we will relate the (mathbb {C})-AWS to the (mathbb {R})-AWS by way of a real structure, which is a real linear, complex anti-linear involution on a complex vector space. This allows for a commutative relation between the real and complex Gaussian fields and the real and complex abstract Wiener spaces. We will construct specific examples which fall under this framework like the complex Brownian motion, complex Feynman-Kac formula and complex fractional Gaussian fields.

We show that, after suitably adjusting a uniform transverse magnetic field, the generic inhomogeneous open XX spin chain has a two-fold degeneracy, and an exact su(2) symmetry whose “inhomogeneous” nonlocal generators depend on coefficients that can be explicitly computed for models associated with discrete orthogonal polynomials.

We show that the i-dimensional plaquette random-cluster model with coefficients in (mathbb {Z}_q) is dual to a ((d-i))-dimensional plaquette random cluster model. In addition, we explore boundary conditions, infinite volume limits, and uniqueness for these models. For previously known results, we provide new proofs that rely more on the tools of algebraic topology.

O(n) loop-decorated random planar maps are a well-studied model coupling quantum gravity with statistical mechanics. An important progress in the study of their geometry was made by Borot, Bouttier and Guitter when they established that O(n) loop-decorated maps could be decomposed recursively by cutting the configurations along the loops. The central building block in this decomposition, called the gasket, is obtained by removing the outermost loops and their interiors. They discovered that for (n<2), at criticality, the gaskets are random maps with high degrees, usually called stable maps. However the case (n=2) remained excluded. We prove that for (n=2) the gaskets of critical rigid O(n) loop-decorated random planar maps are 3/2-stable maps. The case (n=2) thus corresponds to the critical case in random planar maps. Contrary to the cases (n<2), a slowly varying function arises in the perimeter exponent. The proof relies on the Wiener–Hopf factorisation for random walks and is robust enough to deal with bipartite maps with arbitrarily large degrees. Our techniques also provide a characterisation of weight sequences of critical O(2) loop-decorated maps.

Ewens’ sampling formula (ESF) provides the probability distribution governing the number of distinct genetic types and their respective frequencies at a selectively neutral locus under the infinitely-many-alleles model of mutation. A natural and significant question arises: “Is the Ewens probability distribution on regular trees Gibbsian or non-Gibbsian?” In this paper, we demonstrate that Ewens probability distributions can be regarded as non-Gibbsian distributions on regular trees and derive a sufficient condition for the consistency condition. This study lays the groundwork for a new direction in the theory of non-Gibbsian probability distributions on trees.

In this paper, we study the buckling problem of the drifting Laplacian on bounded domains in a complete Riemannian manifold whose corresponding smooth metric measure space has a nonnegative (infty )-dimensional Bakry-Émery Ricci curvature, and then establish some universal inequalities. In particular, our results can reveal the relationship between the ((k+1))-th eigenvalue and the first k eigenvalues relatively quickly, and some methods used in this paper might be applied to other eigenvalue problems.

This paper studies the gradient flow lines for the (L^2) norm square of the Higgs field defined on the moduli space of semistable rank 2 Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface X. The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of X into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram’s resolution of secant varieties.

The purpose of this paper is to bridge the gap between the Dbar method and the direct linearization approach for the lattice Korteweg-de Vries (KdV) type equations. We develop the Dbar method to study some discrete integrable equations in the Adler-Bobenko-Suris list. A Dbar problem is considered to define the eigenfunctions of the Lax pair of the lattice potential KdV equation. We show how an extra parameter is introduced in this approach so that the lattice potential modified KdV equation and lattice Schwarzian KdV equation are derived. We also explain how the so-called spectral Wronskians make sense in constructing the H3((delta )), Q1((delta )) and Q3((delta )) equations. Explicit formulae of multi-soliton solutions are given for the derived equations, from which one can see the connections between the direct linearization variables ((S^{(i,j)}) and V(p)) and the eigenfunctions and their expansions respectively at infinity and a finite point.