Currently I am reading the paper 'Excited Random Walk in One Dimension.'
At page $8$ left column, the authors obtain the following:
Probability that the walk eats precisely $r > 0$ consecutive cookies (we term this event a single “meal”) from the right edge of the cookie-free region is $$P(r) = 2q \frac{\Gamma(L)}{\Gamma(L-2q)} \frac{\Gamma(L+r-1-2q)}{\Gamma(L+r)}$$ where $L-2$ refers to cookie-free gap and $p$ refers to probability of the walk moving to the right and $q$ is the probability of the walk moving to the left.
However, when they calculate the average relative number of consecutive cookies eaten from the right side of the gap, they compute $$\int_0^\infty \tilde{r} \tilde{P}(\tilde{r})\,d\tilde{r}$$ where $\tilde{r} = \frac{r}{L}$ and $\tilde{P} = LP(r).$
Question: Why do they integrate with respect to $\tilde{r}$ with integrand $\tilde{P}?$ I thought to find the average number of cookie eaten, one just needs to compute $$\int_0^\infty r P(r)\, dr$$ instead of the above.
The two expressions denote a change of variable (change of scale): they apply $$ \eqalign{ & 1 = \int_{\,r\, = \,0}^{\;\infty } {P(r)dr} = \int_{\,r\, = \,0}^{\;\infty } {LP(r)d\left( {{r \over L}} \right)} = \cr & = \int_{\,r\, = \,0}^{\;\infty } {LP\left( {L{r \over L}} \right)d\left( {{r \over L}} \right)} = \int_{\,\tilde r\, = \,0}^{\;\infty } {LP\left( {L\tilde r} \right)d\tilde r} = \cr & = \int_{\,\tilde r\, = \,0}^{\;\infty } {\tilde P\left( {\tilde r} \right)d\tilde r} \cr} $$ to pass from $r,P(r)$ to $\tilde r,\tilde P\left( {\tilde r} \right)$ by putting $$ \left\{ \matrix{ \tilde r = r/L \hfill \cr \tilde P\left( {\tilde r} \right) = LP\left( {L\tilde r} \right) = LP\left( r \right) \hfill \cr} \right. $$
This is a totally licit and very common operation done in probability, for example when reconducing a Normal distribution with a given $\sigma$ to the standard one.
They explain that such a "standardization" allows to simplify (in some cases) the expressions by "absorbing" the $L$ parameter, which is in fact a scale parameter. The parallel with the Normal helps to understand why.
That premised, concerning your doubt on the average, $$ \int_{\,\tilde r\, = \,0}^{\;\infty } {\tilde r\,\tilde P\left( {\tilde r} \right)d\tilde r} $$ gives of course the average of $\tilde r$ , denoted as $ \left\langle {\tilde r} \right\rangle$ which tied to $ \left\langle {r} \right\rangle$ by $$ \left\langle {\tilde r} \right\rangle = \left\langle {r/L} \right\rangle = \left\langle r \right\rangle /L $$
In fact, soon after eq. (31) they speak of avg.$\tilde r$ as the "average relative number of consecutive cookies ..": relative is understood to refer to $/L$, and actually immediately below they give $\left\langle {\tilde r} \right\rangle = \left\langle {r/L} \right\rangle = \cdots $.
Addendum
Going back to eq.(30) reported at the beginning of your post $$ P(r) = 2q{{\Gamma (L)} \over {\Gamma (L - 2q)}}{{\Gamma (L + r - 1 - 2q)} \over {\Gamma (L + r)}} $$
The average number of $r$ would be given by $$ \left\langle r \right\rangle = \sum\limits_{0\, < \,r} {r\,P(r)} = 2q{{\Gamma (L)} \over {\Gamma (L - 2q)}}\sum\limits_{0\, \le \,r} {\left( {r + 1} \right)\,{{\Gamma (L + r - 2q)} \over {\Gamma (L + r + 1)}}} $$
In the above $q$ is a real number in the range $(0,1)$; the sum above can be expressed by means of the Gaussian Hypergeometric Function as $$ \left\langle r \right\rangle = {{2q} \over L}\;{}_2F_{\,1} \left( {2,\,L - 2q\,;\;L + 1\,;1} \right) $$ which, in virtue of the Gaussian theorem gives simply $$ \eqalign{ & \left\langle r \right\rangle = {{2q} \over L}{{\Gamma (L + 1)\Gamma ( - 1 + 2q)} \over {\Gamma (L - 1)\Gamma (1 + 2q)}} \quad \left| {\,0 < {\mathop{\rm Re}\nolimits} \left( { - 1 + 2q} \right)} \right.\quad = \cr & = \left\{ {\matrix{ {{{\left( {L - 1} \right)} \over {\left( {2q - 1} \right)}}} & {\left| \matrix{ \;1 \le L \hfill \cr \;1/2 < q \hfill \cr} \right.} \cr \infty & {\left| \matrix{\;1 \le L \hfill \cr \;q \le 1/2 \hfill \cr} \right.} \cr } } \right. \cr } $$ which, for large $L$, correspond to eq.(32).
To this regard we shall note that:
- the summand $\left( {r + 1} \right)\,{{\Gamma (L + r - 2q)} \over {\Gamma (L + r + 1)}}$ has a series expansion at $r=\infty$ which is $1/r^{2q} + O(1/r^{2q+1})$ and the sum is therefore convergent for $1<2q$;
- the Hypergeometric $ {}_2F_{\,1} \left( {a,\,b\,;\;c\,;z} \right)$ has a singularity at $z=1$, so that its value there shall be taken in the limit with due restrictions;
- the restrictions are those provided for the validity of its conversion into the fraction with Gammas, i.e. $0<1-2q$.