Mathematical Physics, Analysis and Geometry最新文献

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.

In this article, we study geometric and analytical features of complete expanding gradient weighted Yamabe solitons, which are self-similar solutions of the weighted Yamabe flow. We show that if the weighted scalar curvature is greater than the soliton constant by an arbitrarily small positive constant, then the soliton is trivial. In contrast, if the weighted scalar curvature is less than the soliton constant by an arbitrarily small positive constant, then the soliton is rotationally symmetric. In addition, we completely classify nontrivial complete expanding gradient weighted Yamabe soliton under some assumed conditions.

We study q-deformation of probability measures on partitions, i.e., q-deformed random partitions. We in particular consider the q-Plancherel measure and show a determinantal formula for the correlation function using a q-deformation of the discrete Bessel kernel. We also investigate Riemann–Hilbert problems associated with the corresponding orthogonal polynomials and obtain q-Painlevé equations from the q-difference Lax formalism.

We introduce a framework to identify Fluctuation Relations for vector-valued observables in physical systems evolving through a stochastic dynamics. These relations arise from the particular structure of a suitable entropic functional and are induced by transformations in trajectory space that are invertible but are not involutions, typical examples being spatial rotations and translations. In doing so, we recover as particular cases results known in the literature as isometric fluctuation relations or spatial fluctuation relations and moreover we provide a recipe to find new ones. We mainly discuss two case studies, namely stochastic processes described by a canonical path probability and non degenerate diffusion processes. In both cases we provide sufficient conditions for the fluctuation relations to hold, considering either finite time or asymptotically large times.