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We investigate two closely related setups. In the first one we consider aTASEP-style system of particles with specified initial and finalconfigurations. The probability of each history of the system is assumed to beequal. We show that the rescaled trajectory of the \emph{second class particle}converges (as the size of the system tends to infinity) to a random arc of anellipse.  In the second setup we consider a uniformly random Young tableau of squareshape and look for typical (in the sense of probability) sliding paths andevacuation paths in the asymptotic setting as the size of the square tends toinfinity. We show that the probability distribution of such paths converges toa random meridian connecting the opposite corners of the square. We alsodiscuss analogous results for non-square Young tableaux.

Second class particles and limit shapes of evacuation and sliding paths for random tableaux