A *Catalan path* of semilength $n$ is a path from $(0,0)$ to $(2n,0)$ that proceeds by taking northeast (1,1) or southeast (1,-1) steps, and never goes below the $x$-axis. The *area* of a path $P$ is the area beneath the path and above the $x$-axis. So, for example, the path of semilength 3 that goes $(0,0)-(1,1)-(2,2)-(3,1)-(4,0)-(5,1)-(6,0)$ has area $5$.


The sum of the area of all Catalan paths of semilength $n$ is known to be $4^n - \binom{2n+1}{n}$, so that the *average* area of a path is on the order of $n^{3/2}$.


In an application I'm working on, I've found that it would be useful to have upper bounds on how many paths have area significantly greater than $n^{3/2}$. The largest possible area is $n^2$, and from Markov's inequality I can say that a vanishing proportion of paths have area greater than $n^2/1000$, say; specifically, there are no more than $O\left(4^n/n^2\right)$ such paths (whereas there are $\Theta\left(4^n/n^{3/2}\right)$ paths in total). 


I suspect that there are far fewer paths with large area, and for my application I need a much lower upper bound. Are there quantitative results for this problem?


I can translate the problem into that of estimating the number of partitions of the number $N$ into at most $c\sqrt{N}$ parts each of size at most $c\sqrt{N}$, which in turn becomes a problem of estimating a coefficient of a Gaussian polynomial, but I'm not aware of quantatitive results here, either. 



Distribution of the area statistic for Catalan paths
Notice that the number of Catalan paths of area at least $cn^{\frac{3}{2}+\varepsilon}$ is less than the number of all paths that deviate from the horizontal axis by at least $n^{\frac{1}{2}+\varepsilon}$. Let $(S\_k, k\geq 0)$ be the one-dimensional simple random walk starting from the origin. We need to find an upper bound for the following probability: 
$$P\big[\max\_{k\leq 2n}S\_k > n^{\frac{1}{2}+\varepsilon}\big].$$


Now, one may use e.g. a Chernoff's bound for the Binomial distribution (together with the Reflection Principle if you don't want an extra polynomial term in front, see e.g. Chapter III of the Feller's book) to obtain that the above probability is upper bounded by $\exp(-c'n^{\varepsilon})$. So, the total number of such paths is at most $4^n\exp(-c'n^{\varepsilon})$.


