Midpoint Distribution of Directed Polymers in the Stationary Regime: Exact Result Through Linear Response

Abstract

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 (J Stat Phys 150:908–939, 2013), thereby rendering our result explicit. In the large length limit that implies that the pdf is nothing but the scaling function $f_{\mathrm{KPZ}}(y)$ introduced by Prähofer and Spohn (J Stat Phys 115(1):255–279, 2004). Using the KPZ-universality paradigm, we find that this function can therefore also be interpreted as the pdf of the position y of the maximum of the Airy process minus a parabola and a two-sided Brownian motion. We provide a direct numerical test of the result through simulations of the Log-Gamma polymer.

Appendix: Imamura–Sasamoto Result

In this appendix we recall for completeness the result obtained by Imamura and Sasamoto in [55] for the analytical expression of the Burgers stationary two-point correlation function. We note that another (presumably equivalent) result could be obtained by using the formulas in the mathematically rigorous work [43]. In [55] Imamura and Sasamoto consider the KPZ equation with the convention

$$\begin{aligned} \partial _t h(t,x) = \frac{\lambda }{2} (\partial _x h(t,x))^2 + \nu \partial _x^2 h(t,x)+ \sqrt{D}\xi (t,x) , \end{aligned}$$

(79)

and we will thus take $D=2 \nu =\lambda = 1$ in their result to conform with our conventions; see Eq. (5). Their result is given in terms of the scaling function $g_t(y)$ which is defined as⁷

$$\begin{aligned} g_t(y) = \int _{-\infty }^\infty s^2 \frac{d F_{w=0,t}(s;y)}{ds} ds- \left( \int _{-\infty }^\infty s \frac{d F_{w=0,t}(s;y)}{ds} ds\right) ^2 . \end{aligned}$$

(80)

And for each (t, y), $F_{w=0,t}(s;y)$ is the cumulative distribution function of the fluctuations of the KPZ interface at (t, y): $g_t(y)$ is the variance of the height at (t, y). The expression for $F_{w=0,t}$ is, using their parameters $\alpha$ and $\gamma _t$ are $\alpha =1$ and $\gamma _t=(t/2)^{1/3})$ (see Theorem 2 in [55])

$$\begin{aligned} F_{w=0,t}(s;X) = \frac{d/ds}{\Gamma (1+\gamma _t^{-1} d/ds)} \int _{{\mathbb R}} du e^{-\gamma _t(s-u)} \left( \nu _{w=0,t}(u;X) -\nu ^{(\delta )}_{w=0,t}(u;X)\right) ,\qquad \end{aligned}$$

(81)

with

$$\begin{aligned}&\nu _{w=0,t}(u;X) = \mathrm{Det}(I - A_{-X,X})L_{-X,X}(u) + \mathrm{Det}(I - A_{-X,X} - D_{-X,X}) \nonumber \\&\nu _{w=0,t}^{(\delta )}(u;X) = \mathrm{Det}(I - A_{-X,X}^{(\delta )})L_{-X,X}^{(\delta )}(u) + \mathrm{Det}(I - A_{-X,X}^{(\delta )} - D_{-X,X}^{(\delta )}) , \end{aligned}$$

(82)

and

$$\begin{aligned}&A_{-X,X}(\xi _1 , \xi _2) = C_t(\xi _1) \int _{u}^{\infty } dy Ai_{\Gamma }^{\Gamma }\left( \xi _1 + y ,\frac{1}{\gamma _t} , 1- \frac{X}{\gamma _t} , 1+ \frac{X}{\gamma _t}\right) \nonumber \\&Ai_{\Gamma }^{\Gamma }\left( \xi _2 + y ,\frac{1}{\gamma _t} , 1+ \frac{X}{\gamma _t} , 1-\frac{X}{\gamma _t}\right) \nonumber \\&D_{-X,X}(\xi _1,\xi _2) = (A_{-X,X}C_t B_{-X,X,u})(\xi _1)B_{X,-X,u}(\xi _2) \nonumber \\&L_{X,-X}(u) = - \frac{2\gamma _E}{\gamma _t} + u -X^2 -1 \nonumber \\&\!+\! \int _{{\mathbb R}} dx C_t(x)\left( B_{\!-\!X,X,u}^{(1)}(x) B_{X,\!-\!X,u}^{(2)}(x) \!+\! B_{X,-X,u}^{(1)}(x) B_{-X,X,u}^{(2)}(x) \!-\! B_{-X,X,u}^{(2)}(x) B_{X,-X,u}^{(2)}(x) \right) \nonumber \\&B_{-X,X,u}^{(1)}(x) = e^{-X^3/2 + (x+u) X},\nonumber \\&B_{-X,X,u}^{(2)}(x) = \int _{0}^{\infty } d\lambda e^{-X\lambda } Ai_{\Gamma }^{\Gamma }\left( x + u + \lambda , \frac{1}{\gamma _t} , 1 + \frac{X}{\gamma _t} , 1 - \frac{X}{\gamma _t}\right) \nonumber \\&B_{-X,X,u}(x) = B_{-X,X,u}^{(1)}(x) - B_{-X,X,u}^{(2)}(x), \quad C_t(x) = \frac{e^{\gamma _t x }}{e^{\gamma _t x}-1}. \end{aligned}$$

(83)

The expression with the superscript $\delta$ are identical with $C_t(x) \rightarrow C_t(x)^{(\delta )} = C_t(x) - \delta (x)$. $\gamma _E$ in the expression of $L_{X,-X}(u)$ denotes Euler’s gamma constant. Finally the function $Ai_{\Gamma }^{\Gamma }$ is a deformed Gamma function defined by

$$\begin{aligned} Ai_{\Gamma }^{\Gamma }(a,b,c,d) = \frac{1}{2\pi } \int _{\Gamma _{id/b}} dz e^{i z a + i \frac{z^3}{3}} \frac{\Gamma (ib z + d)}{\Gamma (-ibz + c)} \end{aligned}$$

(84)

with $\Gamma _{id/b}$ a contour from $-\infty$ to $+\infty$ passing below id / b. In (81) $\frac{1}{\Gamma (1 + \gamma _t^{-1} d/ds)}$ is the operator defined by the Taylor expansion of the Gamma function around 1, and $\mathrm{Det}(I-K)$ denotes the Fredholm determinant of the operator with the kernels defined above, and $(\xi _1,\xi _2)\in {\mathbb R}^2$. No matter how complicated this result seems it can indeed be plotted, see Fig. 3 in [55]. In the limit $t \rightarrow \infty$ the results simplifies a bit. In that case $g_{\infty }(y)$ is still given by a variance, but now associated with the cumulative distribution function $F_{w=0}(s;y)=\lim _{t\rightarrow \infty }F_{x=0,t}(s;y)$. An explicit expression for $F_{w=0}(s;y)$ was given in Theorem 3 of [55].

Let us finally make the connection with our setting. In [55] $g_t(y)$ is related to Burgers’ stationary two-point correlation through (see around Corollary 4)

$$\begin{aligned} g_t(y)= \left( \frac{2}{ t} \right) ^{2/3} C(t , (2 t^2)^{1/3} y) , \end{aligned}$$

(85)

and

$$\begin{aligned} \langle u(t,x) u(0,0) \rangle _0^0 = \frac{1}{2} \partial _x^2 C(t,x) . \end{aligned}$$

(86)

Following our result (8) and the rescaling (9) one thus easily arrives at Eq. (10).