One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

Stochastic equations in infinite dimensions

The Bootstrap and Edgeworth Expansion

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $$\Phi ^4_3$$
 Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of $$3$$
 -dimensional ferromagnets near their critical temperature.

/pdf/a-theory-of-regularity-structures-58w6z3cq21.pdf

A theory of regularity structures

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/85808/1/KJ00004746786.pdf

Time-Dependent Statistics of the Ising Model

We consider ASEP on a bounded interval and on a half-line with sources and sinks. On the full line, Bertini and Giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the KPZ equation. We prove here that under similar weakly asymmetric scaling of the sources and sinks as well, the bounded interval ASEP height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides (different parameter for each side), and likewise for the half-line ASEP to KPZ on a half-line. This result can be interpreted as showing that the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP.© 2018 Wiley Periodicals, Inc.

Open ASEP in the Weakly Asymmetric Regime

We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the discrete time geometric $$q$$
 -TASEP model, studied earlier by Borodin and Corwin, scales to this polymer model in the limit $$q\rightarrow 1$$
 . This allows us to exploit the exact results for geometric $$q$$
 -TASEP to derive a Fredholm determinant formula for the strict-weak polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling and GUE Tracy–Widom limit for the free energy fluctuations. We also derive moments formulae for the polymer partition function directly by Bethe ansatz, and identify the limit of the free energy using a stationary version of the polymer model.

/pdf/the-strict-weak-lattice-polymer-ues4yeeoud.pdf

The Strict-Weak Lattice Polymer

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter $\beta$, which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when $\beta^2 \in (0,\frac{16\pi}{3})$ the Wick renormalised equation is well-posed. In the regime $\beta^2 \in (0,4\pi)$, the Da Prato-Debussche method applies, while for $\beta^2 \in [4\pi,\frac{16\pi}{3})$, the solution theory is provided via the theory of regularity structures (Hairer 2013). We also show that this model arises naturally from a class of $2+1$-dimensional equilibrium interface fluctuation models with periodic nonlinearities. 
The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed by Lacoin, Rhodes and Vargas (2013).

The dynamical sine-Gordon model

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter $${\beta}$$
 , which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when $${\beta^{2} \in (0, \frac{16\pi}{3})}$$
 the Wick renormalised equation is well-posed. In the regime $${\beta^{2} \in (0, 4\pi)}$$
 , the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for $${\beta^{2} \in [4\pi, \frac{16\pi}{3})}$$
 , the solution theory is provided via the theory of regularity structures [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of $${2 + 1}$$
 -dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al. [Commun Math Phys 337(2):569–632, 2015].

/pdf/the-dynamical-sine-gordon-model-4676qlpyyf.pdf

The Dynamical Sine-Gordon Model

We consider the KPZ equation in one space dimension driven by a stationary centred space–time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf–Cole solution to the KPZ equation driven by space–time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.

/pdf/a-central-limit-theorem-for-the-kpz-equation-427nm5pjxw.pdf

Hao Shen

Papers

Open ASEP in the Weakly Asymmetric Regime

The Strict-Weak Lattice Polymer

The dynamical sine-Gordon model

The Dynamical Sine-Gordon Model

A central limit theorem for the KPZ equation