From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions

We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of i.i.d.variables in 1 + 1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al. (2014), we show that the scaled partition function, as a process defined on [0, 1] x R, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Ito integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved. (C) 2019 Published by Elsevier B.V.