content_image/397919.jpg content_image/397920.jpg content_image/397921.jpg content_image/397928.jpg content_image/397918.jpg content_image/397924.jpg content_image/397926.jpg content_image/397922.jpg
We rearrange the integration limits, as explained in Fig. 8 

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Hence we can use 

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Repeating this procedure we rewrite Eq. (94) as 

<img src='content_image/397921.jpg'>

<img src='content_image/397928.jpg'>

where Nor is a normalization constant and Z is defined in Eq. (17). 

Thus Eq. (97) describes a problem of order statistics which is extensively investigated by mathematicians, as mentioned in Sec. III D. P $^{eq}$( x$_{T}$ ) Eq. (97) is the PDF of the random variable which has exactly N random vari- able larger than it and N smaller. In this sense we have transformed the problem to a non-interacting system, similar to the non interacting picture in the main text. The information contained in the single non-interacting particle, i.e. the single particle Boltzmann distribution exp[ − V ( x ) /k$_{b}$T ] /Z is all what is needed for the calcula- tion of the position of the tagged particle. 

Using the symmetry V ( x ) = V ( − x ) we have 

[1] T. M. Ligget Interacting Particle Systems (springer- Verlag, New York 1985). 

[2] B. Derrida, E. Domany and D. Mukamel, J. of Statistical Physics 69 667 (1992). 

[3] B. Derrida Phys. Rep. 301 , 65 (1998). 

[4] T. E. Harris, J. Appl. Probab. 2 , 323, (1965). 

<img src='content_image/397918.jpg'>

and 

x$_{T}$ 0 Using Eqs. (98,99), we rewrite Eq. (97) 

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In the limit N → ∞ only x$_{T}$ with ∫ x$_{T}$ 0 exp[ − V ( x ) /k$_{b}$T ]d x/Z ≪ 1 will have a measur- able contribution to P $^{eq}$( x$_{T}$ ), since if this condition is not satisfied, the value of P $^{eq}$( x$_{T}$ ) is exponentially small in N . Expanding the ln in Eq. (100) we have 

<img src='content_image/397926.jpg'>

Since N >> 1 we expand ∫ 0 exp( − V ( x ) /k$_{b}$T )d x = x$_{T}$ where we used V (0) = 0. We use V ( x$_{T}$ ) /k$_{b}$T ≪ N ( x$_{T}$ ) $^{2}$/Z2 which holds in the center part of the PDF of the tagged particle (since N >> 1, V (0) = 0, and V ( x ) is analytic) P $^{eq}$( x$_{T}$ ) ∼ C exp [ − 4 N ( x$_{T}$ ) 2 Z 2 ] where C is a normalization constant. This final approximate re- sult is the same as Eq. (21), justifying the tricks used to derive our main results. 

[5] D. W. Jepsen, J. Math. Phys. (N.Y.) 6 , 405 (1965). 

[6] D. G. Levitt, Phys. Rev. A 8 , 3050 (1973). 

[7] S. Alexander and P. Pincus, Phys. Rev. B 18 , 2011 (1978); P. A. Fedders, Phys. Rev. B 17 , 40 (1978); H. van Beijeren, K. W. Kehr, and R. Kutner Phys. Rev. B 28 , 5711 (1983). S. Gupta, S. N. Majumdar, C. Go- 

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