The intermediate disorder regime for Brownian directed polymers in Poisson environment

We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime (Alberts et al., 2014), which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener–Itô chaos expansion (Last and Penrose, 2017), which, applied to the partition function, is the basic ingredient of the proof.     

On the long time behavior of the stochastic heat equation

We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum – the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process. Received: 11 November 1997 / Revised version: 31 July 1998     

On the partition function of a directed polymer in a Gaussian random environment

 The purpose of this work is the study of the partition function of a -dimensional lattice directed polymer in a Gaussian random environment being the inverse of temperature). In the low-dimensional cases , we prove that for all , the renormalized partition function converges to 0 and the correlation of two independent configurations does not converge to 0. In the high dimensional case (), a lower tail of has been obtained for small . Furthermore, we express some thermodynamic quantities in terms of the path measure alone. Received: 8 June 2001 / Revised version: 8 February 2002 / Published online: 22 August 2002 Mathematics Subject Classification (2000): 60K37, 82D30 Key words or phrases: Directed polymer in random environment – Gaussian environment – partition function     

A mixed-step algorithm for the approximation of the stationary regime of a diffusion

In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called the mixed-step scheme, where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson–Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we conclude by giving sufficient “asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.     

Sharp critical behavior for pinning models in a random correlated environment

This article investigates the effect for random pinning models of long range power-law decaying correlations in the environment. For a particular type of environment based on a renewal construction, we are able to sharply describe the phase transition from the delocalized phase to the localized one, giving the critical exponent for the (quenched) free-energy, and proving that at the critical point the trajectories are fully delocalized. These results contrast with what happens both for the pure model (i.e., without disorder) and for the widely studied case of i.i.d. disorder, where the relevance or irrelevance of disorder on the critical properties is decided via the so-called Harris Criterion (Harris, 1974) [21].     

High temperature regime for a multidimensional Sherrington–Kirkpatrick model of spin glass

Comets and Neveu have initiated in [5] a method to prove convergence of the partition function of disordered systems to a log-normal random variable in the high temperature regime by means of stochastic calculus. We generalize their approach to a multidimensional Sherrington-Kirkpatrick model with an application to the Heisenberg model of uniform spins on a sphere of ℝ ^d , see [9]. The main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path. Received: 30 October 1996 / In revised form: 13 October 1997     

Vectors of two-parameter Poisson-Dirichlet processes

The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson-Dirichlet processes. It is well-known that each component can be obtained by resorting to a change of measure of a σ-stable process. Thus dependence is achieved by applying a Lévy copula to the marginal intensities. In a two-sample problem, we determine the corresponding partition probability function which turns out to be partially exchangeable. Moreover, we evaluate predictive and posterior distributions.     

Properties of 1D Classical and Quantum Ising Models: Rigorous Results

In this paper, we consider one-dimensional classical and quantum spin-1/2 quasi-periodic Ising chains, with two-valued nearest neighbor interaction modulated by a Fibonacci substitution sequence on two letters. In the quantum case, we investigate the energy spectrum of the Ising Hamiltonian, in presence of constant transverse magnetic field. In the classical case, we investigate and prove analyticity of the free energy function when the magnetic field, together with interaction strength couplings, is modulated by the same Fibonacci substitution (thus proving absence of phase transitions of any order at finite temperature). We also investigate the distribution of Lee–Yang zeros of the partition function in the complex magnetic field regime, and prove its Cantor set structure (together with some additional qualitative properties), thus providing a rigorous justification for the observations in some previous works. In both, quantum and classical models, we concentrate on the ferromagnetic class.     

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.     

d维Bernstein算子加Jacobi权的收敛阶

１．引 言 设Ｓ＝Ｓｄ（ｄ＝１,２,…）是 Ｒｄ中的单纯形,即记ｋ＝（ｋ１,ｋ２,……,ｋｄ）∈Ｒｄ,ｋｉ为非负整数, ,则Ｓ上定义的函数ｆ所对应的ｄ维Ｂｅｒｎｓｔｅｉｎ算子定义为其中 Ｐｎ,ｋ（ｘ）＝是 Ｂｅｒｎｓｔｅｉｎ基函数．引进多维Ｊａｃｏｂｉ权函数, 这里 ．定义Ｂｅｒｎｓｔｅｉｎ权函数 表示微分算子． 记 是单位向量,即第ｉ个分量为１,其余ｄ－１个分量为０, ．定义函数ｆ在方向ｅ上的ｒ阶对称差分为Ｃ（Ｓ）中的加权Ｓｏｂｏｌｅｖ空间为其中Ｓ为Ｓ的内部．定义加权Ｋ－泛函及加权光滑模其中 为加权范数． …     

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.     

The non-emptiness of the core of a partition function form game

The purpose of this paper is to provide a necessary and sufficient condition for the non-emptiness of the core for partition function form games. We generalize the Bondareva–Shapley condition to partition function form games and present the condition for the non-emptiness of “the pessimistic core”, and “the optimistic core”. The pessimistic (optimistic) core describes the stability in assuming that players in a deviating coalition anticipate the worst (best) reaction from the other players. In addition, we define two other notions of the core based on exogenous partitions. The balanced collections in partition function form games and some economic applications are also provided.     

Transport equation with boundary conditions for free surface localization

Summary. During the filling stage of an injection moulding process, which consists in casting a melt polymer in order to manufacture plastic pieces, the free interface between polymer and air has to be precisely described. We set this interface as a zero level set of an unknown function. This function satisfies a transport equation with boundary conditions, where the velocity field has few regularity properties. In a first part, we obtain existence and uniqueness result for these equations, under weaker regularity assumptions than C. Bardos [Bar70], and C. Bardos, Y. Leroux and J.C. Nedelec [BLN79] in previous articles, but stronger assumptions than R.J. DiPerna and P.L. Lions [DL89b] who studied the case without boundary condition. We also study some regularity properties of the interface. A second part is devoted to an application to injection molding of melt polymer. We give a numerical experiment which shows that our method leads to an accurate localization of interface, which is robust, since it easily handles changes of topology of the free interface, as bubble formation or fusion of two fronts of melt polymer. Received November 1, 1997 / Revised version received December 9, 1998 / Published online September 24, 1999     

On convergence determining and separating classes of functions

Herein, we generalize and extend some standard results on the separation and convergence of probability measures. We use homeomorphism-based methods and work on incomplete metric spaces, Skorokhod spaces, Lusin spaces or general topological spaces. Our contributions are twofold: we dramatically simplify the proofs of several basic results in weak convergence theory and, concurrently, extend these results to apply more immediately in a number of settings, including on Lusin spaces.     

Identities and congruences for the general partition and Ramanujan’s tau functions

We present some identities and congruences for the general partition function p _r (n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate identities and congruences for general partition function.     

Stiff Oscillatory Systems, Delta Jumps and White Noise

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is <e6>OM</e6>(N). The model problems are integrated numerically in the stiff regime where the time-step satisfies The convergence of the algorithms is studied in this case in the limit and For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically. August 25, 1999. Final version received: May 3, 2000.     

Classical spin lattice systems with vacuum

We present cluster properties for the lattice spin systems with general n-body interaction and we apply it to find the asymptotic expansion of the logarithm of the partition function in powers of the volume as well as a local limit theorem for the particle number.     

An adaptive scheme for the approximation of dissipative systems

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, a regular explicit Euler scheme–with constant or decreasing step–may explode and implicit Euler schemes are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.