Thesis title: Phase transition for the Stochastic Sandpile model and cutoff for the East model
This thesis focuses on two interacting particle systems with underlying physical motivation. The Stochastic Sandpile model (SSM) was introduced to study self-organized criticality, while the East model is one of the most studied Kinetically Constrained models (KCM), a class introduced to model the liquid–glass transition.
The SSM is a continuous-time particle system on a graph, where each vertex is considered stable if it contains at most one particle and unstable otherwise. Initially, each vertex has a number of particles drawn independently from a Poisson distribution with mean
$\mu>0$. Each unstable site topples at rate 1, sending two particles to neighbouring sites, each moving independently and choosing its destination uniformly at random.
There are two versions of the SSM: the fixed-energy version, in which the total number of particles is conserved, and the driven-dissipative version, in which particles are added at each step and may leave the system during stabilisation.
The fixed-energy version of SSM on infinite graphs exhibits a phase transition: when the particle density $\mu$ is low, the system locally fixates, while for high $\mu$ it remains active. One
of the key questions for this model is whether it has a non-trivial phase transition, with a critical density strictly between zero and one. We show that the critical density of the model on $\Z^d$ is strictly less than one for any dimension $d\geq 1$, generalising a previous result
which was limited to the one-dimensional case. In addition, we show that the critical density is strictly positive on any infinite vertex-transitive graph, extending previous results
and providing a simpler proof.
We also consider the driven-dissipative version of SSM on finite graphs. First, we describe a procedure to exactly sample from the stationary
distribution of the model in all connected finite graphs.
Then, we study the model on
the complete graph with a number of vertices tending to infinity and show that the
stationary density tends to $1/2$.
KCM are spin systems in which each site of a graph can be infected (state "0") or healthy (state "1"), with infections representing facilitating states that allow updates at neighbouring sites. Each site tries at rate 1 to update its state, and the update is accepted only if a specific constraint, determined by the configuration in the neighbourhood, is satisfied.
In the East model on $\Z^d$, the constraint for the update at $x \in \Z^d$ requires an infection between sites $x-\vec e_i$ for some $\vec e_i$ in the canonical basis of $\Z^d$.
We study the East model on $\Z^d$ together with one of its natural variant, in which the updating rate is proportional to the number of infected neighbours, which we refer to as the Modified East model. Under any ergodic boundary condition it is known that the mixing time of the East chain in a box of side $L$ is $\Theta(L)$ for any $d\ge 1$.
Moreover, with minimal boundary conditions and at low temperature, i.e., low equilibrium density of the facilitating vertices, the East chain exhibits cutoff around the mixing time of the $d=1$ case.
We extend this result to high equilibrium density of the facilitating vertices. As in the low density case, the key approach is to prove that the speed of infection propagation in the $(1,1,\dots,1)$ direction is larger than
the same speed along a coordinate direction. By borrowing a technique from first passage percolation, the proof links the result to the precise value of the critical probability of oriented (bond or site) percolation on $\Z^d$.