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Register a new userMultilinearity of two-point correlation functions in one-dimensional models out of equilibrium
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In this note we consider non-equilibrium steady states of one-dimensional models of heat conduction (wealth exchange) which are coupled to some reservoirs creating currents. In particular we will give sufficient and necessary conditions which will depend only on the first two moments of the reservoir measures and the redistribution parameter under which the two-point functions are multilinear. This presents the first example of multilinear two-point functions in the absence of product stationary measures.
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We consider long-range self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)asymp|x|^{-d-alpha}$ with $alpha>0$. The upper-critical dimension $d_{mathrm{c}}$ is $2(alphawedge2)$ for self-avoiding walk and the Ising model, and $3(alphawedge2)$ for percolation. Let $alpha e2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{alphawedge2-d}$, where the constant $Cin(0,infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $alpha<2$ and $alpha>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.
We derive and prove exponential and form factor expansions of the row correlation function and the diagonal correlation function of the two dimensional Ising model.
Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with anchored loops, boundaries with SLE processes and with double SLE processes. We focus further upon n=0, representing self-avoiding loops, which corresponds to a logarithmic conformal field theory (LCFT) with c=0. In this limit the twist operator plays the role of a zero weight indicator operator, which we verify by comparison with known examples. Using the additional conditions imposed by the twist operator null-states, we derive a new explicit result for the probabilities that an SLE_{8/3} wind in various ways about two points in the upper half plane, e.g. that the SLE passes to the left of both points. The collection of c=0 logarithmic CFT operators that we use deriving the winding probabilities is novel, highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory. We provide evidence that both partners do appear in the theory, one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk-boundary fusion rules.
An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In particular the functions of such systems satisfy a set of partial differential equations that defines an infinite Lie group. Emphasis is put on the functional transformations of a particular two-dimensional hypercomplex number system, capable of maintaining the wave equation as invariant and then the speed of light invariant too. These functional transformations describe accelerated frames and can be considered as a generalization of two dimensional Lorentz group of special relativity. As a first application the relativistic hyperbolic motion is obtained.
Within the framework of quantum mechanics working with one-dimensional, manifestly non-Hermitian Hamiltonians $H=T+V$ the traditional class of the exactly solvable models with local point interactions $V=V(x)$ is generalized. The consequences of the use of the nonlocal point interactions such that $(V f)(x) = int K(x,s) f(s) ds$ are discussed using the suitably adapted formalism of boundary triplets.
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