A system of two types (\(X\) and \(Y\)) of interacting particles on \(\mathbb{Z}\) is studied. \(X\) particles movement is described by independent, continuous time, symmetric, simple random walk with total jump rate \(D_X=2\) whereas \(Y\) (inert) particles evolve according to a random walk with total jump rate \(D_Y=0\). Initially, there are \(\eta(x)\) of \(X\) particles at sites \(x=0,-1,-2,\dots\) (with \(\eta(x)\geq 1\) for some \(x\)) and a fixed number \(a\) of \(Y\) particles at \(x=1,2,\dots\). When a site \(x=1,2,\dots\) is visited by an \(X\) particle for the first time, all \(Y\) particles located at \(x\) are instantaneously turned into \(X\) particles and start moving. This model can be interpreted as an infection process or as a combustion reaction.
Let \(r_t\) be the rightmost site that has been visited by an \(X\) particle up to time \(t\) (\(r_0:=0\)). Due to F. Comets et al. [Trans. Am. Math. Soc. 361, No. 11, 6165–6189 (2009; Zbl 1177.82081)] if \(\sum_{x\leq 0} \exp(\theta x)\eta(x)<\infty\;(\ast)\) for a small enough \(\theta >0\), then \(r_t/t\to v\) a.s. as \(t\to \infty\), here a constant \(v\in (0,\infty)\). The main result by J. Bérard and A. Ramirez states that if \((\ast)\) is satisfied for all \(\theta >0\), then the large deviation principle holds for \(r_t/t\) (as \(t\to \infty)\) with some rate function \(I\). Moreover, the lower bound is given for \(\mathbb{P}(c\leq r_t/t \leq b)\) for all \(0\leq c<b <v\), the upper bound is provided for \(\mathbb{P}(r_t/t \leq b)\) for each \(0\leq b <v\) and two-sided inequalities are obtained for \(\mathbb{P}(r_t=0)\) as \(t\to \infty\).