Midpoint distribution of directed polymers in the stationary regime: exact result through linear response

Understand

We obtain an exact result for the midpoint probability distribution function (pdf) of the stationary continuum directed polymer, when averaged over the disorder.

• It is obtained by relating that pdf to the linear response of the stochastic Burgers field to some perturbation.
• From the symmetries of the stochastic Burgers equation we derive a fluctuation-dissipation relation so that the pdf gets given by the stationary two space-time points correlation function of the Burgers field.
• An analytical expression for the latter was obtained by Imamura and Sasamoto [2013], thereby rendering our result explicit.