It was shown by Basu, Sidoravicius and Sly that a TASEP starting
with the step initial condition, i.e., with one particle each at every
nonpositive site of Z and no particle at positive sites, with a slow
bond at the origin where a particle jumping from the origin jumps at
a smaller rate r < 1, has an asympototic current which is strictly less
than 1/4. Here we study the limiting measure of the TASEP with a
slow bond. The distribution of regular TASEP started with the step
initial condition converges to the invariant product Bernoulli measure
with density 1/2 . The slowdown due to the slow bond implies that there
is a long range e?ect near the origin where the region to the right of
origin is sparser and there is a tra?c jam to the left of the slow bond
with particle density higher than a half. However, the distribution
becomes close to a product Bernoulli measure as one moves far away
from the origin, albeit with a di?erent density ? < 1/2 to the right of
the origin and ?' > 1/2 to the left of the origin. This answers a question
due to Liggett. The proof uses the correspondence between TASEP
and directed last passage percolation on Z^2 with exponential passage
times, and the geometric properties of the maximal paths there.