Last passage percolation models are fundamental examples in statistical mechanics where the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called {\em geodesics}. Here we consider the Poissonian last passage percolation (LPP) and the exponential directed last passage percolation
(DLPP), the latter having a standard coupling with another classical interacting particle system, the totally asymmetric simple exclusion process or TASEP. These belong to the so-called KPZ universality class, for which exact algebraic formulae have led to precise results for fluctuations and scaling limits. However, such formulae are not very robust and studying the geometry of the geodesics can often provide new insights into these models. Here we consider three problems in each of these three models; exponential DLPP, TASEP and Poissonian LPP, and see how geometric and probabilistic techniques solve such problems.
In the first problem, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. We consider directed last passage percolation on $\Z^2$ with i.i.d. exponential weights on the vertices. Fix two points $v_1=(0,0)$ and $v_2=(0, \lfloor k^{2/3} \rfloor)$ for some $k>0$, and consider the maximal paths $\Gamma_1$ and $\Gamma_2$ starting at $v_1$ and $v_2$ respectively to the point $(n,n)$ for $n\gg k$. Our object of study is the point of coalescence, i.e., the point $v\in \Gamma_1\cap \Gamma_2$ with smallest $|v|_1$. We establish that the distance to coalescence $|v|_1$ scales as $k$, by showing the upper tail bound $\P(|v|_1> Rk) \leq R^{-c}$ for some $c>0$. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction $(1,1)$ starting from $v_3=(-\lfloor k^{2/3} \rfloor , \lfloor k^{2/3}\rfloor)$ and $v_4=(\lfloor k^{2/3} \rfloor ,- \lfloor k^{2/3}\rfloor)$, we establish the optimal tail estimate $\P(|v|_1> Rk) \asymp R^{-2/3}$, for the point of coalescence $v$. This answers a question left open by Pimentel(2016) who proved the corresponding lower bound.
Next, we study the ``slow bond" model, where the totally asymmetric simple exclusion process (TASEP) on $\Z$ is modified by adding a slow bond at the origin. The slow bond increases the particle density immediately to its left and decreases the particle density immediately to its right. Whether or not this effect is detectable in the macroscopic current started from the step initial condition has attracted much interest over the years and this question was settled recently in Basu-Sidoravicius-Sly (2014) where it was shown that the current is reduced even for arbitrarily small strength of the defect. Following non-rigorous physics arguments in Janowsky-Lebowitz(1992,1994) and some unpublished works by Bramson, a conjectural description of properties of invariant measures of TASEP with a slow bond at the origin was provided by Liggett in his book (1999). We establish Liggett's conjectures and in particular show that, starting from step initial condition, TASEP with a slow bond at the origin, as a Markov process, converges in law to an invariant measure that is asymptotically close to product measures with different densities far away from the origin towards left and right. Our proof exploits the correspondence between TASEP and the last passage percolation on $\Z^2$ with exponential weights and uses the understanding of geometry of maximal paths in those models.
Finally, we study the modulus of continuity of polymer fluctuations and weight profiles in Poissonian LPP. The geodesics and their energy in Poissonian LPP can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order, and we refer to scaled geodesics
as {\em polymers} and
their scaled energies as {\em weights}. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3}\big(\log t^{-1}\big)^{1/3}$.
The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation $t$ (and a horizontal separation of the same order),
the maximum transversal fluctuation has order $t^{2/3}\big(\log t^{-1}\big)^{1/3}$.
Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$, $z\in [1,2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3}\big(\log t^{-1}\big)^{2/3}$.
In this way, we identify exponent pairs of $(2/3,1/3)$
and $(1/3,2/3)$ in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation.
The two exponent pairs describe [Hammond(2012, 2012, 2011)] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.
