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Register a new userIntegral formulas of ASEP and $q$-TAZRP on a ring
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In this paper, we obtain the transition probability formulas for the Asymmetric Simple Exclusion Process (ASEP) and the $q$-deformed Totally Asymmetric Zero Range Process ($q$-TAZRP) on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition.
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In [AAV] Amir, Angel and Valk{o} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-ty
We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms.
We construct, for the first time to our knowledge, a one-dimensional stochastic field ${u(x)}_{xin mathbb{R}}$ which satisfies the following axioms which are at the core of the phenomenology of turbulence mainly due to Kolmogorov: (i) Homogeneity and isotropy: $u(x) overset{mathrm{law}}= -u(x) overset{mathrm{law}}=u(0)$ (ii) Negative skewness (i.e. the $4/5^{mbox{tiny th}}$-law): $mathbb{E}{(u(x+ell)-u(x))^3} sim_{ell to 0+} - C , ell,,$ , for some constant $C>0$ (iii) Intermittency: $mathbb{E}{|u(x+ell)-u(x) |^q} asymp_{ell to 0} |ell|^{xi_q},,$ for some non-linear spectrum $qmapsto xi_q$ Since then, it has been a challenging problem to combine axiom (ii) with axiom (iii) (especially for Hurst indexes of interest in turbulence, namely $H<1/2$). In order to achieve simultaneously both axioms, we disturb with two ingredients a underlying fractional Gaussian field of parameter $Happrox frac 1 3 $. The first ingredient is an independent Gaussian multiplicative chaos (GMC) of parameter $gamma$ that mimics the intermittent, i.e. multifractal, nature of the fluctuations. The second one is a field that correlates in an intricate way the fractional component and the GMC without additional parameters, a necessary inter-dependence in order to reproduce the asymmetrical, i.e. skewed, nature of the probability laws at small scales.
We obtain a dimensional reduction result for the law of a class of stochastic differential equations using a supersymmetric representation first introduced by Parisi and Sourlas.
This paper applies the phase-integral method to the stationary theory of alpha-decay. The rigorous form of the connection formulae, and their one-directional nature that was not widely known in the physical literature, are applied. The condition for obtaining s-wave metastable states affects the stationary state at large distance from the nucleus, which is dominated by the cosine of the phase integral minus (pi over 4). Accurate predictions for the lowest s-wave metastable state and mean life of the radioactive nucleus are obtained in the case of Uranium. The final part of the paper describes the phase-integral algorithm for evaluating stationary states by means of a suitable choice of freely specifiable base function. Within this framework, an original approximate formula for the phase integrand with arbitrary values of the angular momentum quantum number is obtained.
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