14 We rearrange the integration limits, as explained in Fig. 8 Hence we can use Repeating this procedure we rewrite Eq. (94) as and Using Eqs. (98,99), we rewrite Eq. (97) 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). [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- ∫ x$_{T}$ −∞ $_{e}$− V ( x ) k B T Z d x = 1 2 + ∫ x$_{T}$ 0 $_{e}$− V ( x ) k B T Z d x, (98) ∫ x$_{T}$ −∞ d $^{x}$− N ∫ x$_{T}$ x$_{−}$$_{N}$ d $^{x}$− N +1 · · · = ∫ x$_{T}$ −∞ d $^{x}$− N ∫ x$_{−}$$_{N}$ −∞ d $^{x}$− N +1 · · · . (95) ∫ x$_{T}$ −∞ d $^{x}$− N ∫ x$_{T}$ x$_{−}$$_{N}$ d $^{x}$− N +1 · · · = 1 2 [ ∫ x$_{T}$ −∞ d $^{x}$− N ∫ x$_{T}$ x$_{−}$$_{N}$ d $^{x}$− N +1 · · · + ∫ x$_{T}$ −∞ d $^{x}$− N ∫ x$_{−}$$_{N}$ −∞ d $^{x}$− N +1 · · · ] = 1 2 ∫ x$_{T}$ −∞ d x$_{N}$ ∫ x$_{T}$ −∞ d $^{x}$− N +1 · · · . (96) P $^{eq}$( x$_{T}$ ) = Nor    ∫ x$_{T}$ −∞ d x exp [ − V ( x ) k$_{B}$T ] Z   $^{N}$    ∫ ∞ x$_{T}$ d x exp [ − V ( x ) k$_{B}$T ] Z   $^{N}$ exp [ − V ( x$_{T}$ ) k$_{B}$T ] Z . (97) ∫ ∞ x$_{T}$ $_{e}$− V ( x ) k B T Z d x = 1 2 − ∫ x$_{T}$ 0 $_{e}$− V ( x ) k B T Z d x. (99) 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 P $^{eq}$( x$_{T}$ ) = Nor e − V ( x T ) /k b T Z e N ln 1 4 exp { N ln [ 1 − 4 ( ∫ x T 0 e − V ( x ) /k b $^{T}$d x Z d x ) 2 ]} . (100) Since N >> 1 we expand ∫ x$_{T}$ 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. P $^{eq}$( x$_{T}$ ) ∝ exp [ − V ( x$_{T}$ ) k$_{b}$T ] exp    − 4 N   ∫ x$_{T}$ 0 d $_{xe}$− V ( x ) k b T Z  $^{2}$    . (101) FIG. 8: Left panel integration in the domain −∞ < x − N < x$_{T}$ , x − N < x − N +1 < x$_{T}$ is equivalent to integration −∞ < x − N < x$_{T}$ , −∞ < x − N $_{+1}$< x − N (right panel). GLYPH<MT88>$_{GLYPH<MT45>GLYPH<MT78>GLYPH<MT43>GLYPH<MT49>}$ GLYPH<MT88> GLYPH<MT84> GLYPH<MT88> GLYPH<MT84> GLYPH<MT88>$_{GLYPH<MT45>GLYPH<MT78>}$ GLYPH<MT88> GLYPH<MT84> GLYPH<MT88> GLYPH<MT84> GLYPH<MT88>$_{GLYPH<MT45>GLYPH<MT78>}$ GLYPH<MT88>$_{GLYPH<MT45>GLYPH<MT78>GLYPH<MT43>GLYPH<MT49>}$