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For independent site percolation or bond percolation on a rooted locally finite graph, $\theta (p)$ is right continuous on $[0,1)$. Finite-radius connection events have polynomial probabilities in $p$ and decrease to infinite connection. Thus $\theta$ is their infimum by continuity from above of a measure. This infimum of continuous functions is upper semicontinuous; parameter monotonicity from the monotone coupling of Bernoulli percolation then gives right continuity.

A simple random walk on a locally finite graph chooses each vertex in the current vertex's graph neighbourhood with equal probability, assuming there is at least one available neighbour. On the integer line there are two choices, one unit left or right, so each has probability $1/2$. This is the one-dimensional simple symmetric random walk. Terminology varies: a simple random walk on the integer line is also often allowed unequal step probabilities, with “symmetric” specifying the fair case.

For a simple symmetric random walk starting at $S_0=0$, each increment has the Rademacher distribution, with expected value zero and variance one. Thus linearity of expectation and the variance of a sum of independent random variables giveFor example, after $80$ steps the expected value is zero, the variance is $80$, and the standard deviation is $\sqrt{80}\approx 8.944$. Doubling the typical distance scale takes four times as many steps. This square-root growth explains why a random walk can wander far even when there is no preferred direction.

The colored curves are illustrative sample paths, not averages over all possible walks. The dashed curves show the expected values; the shaded bands show the expected value plus or minus one standard deviation at each time. These bands are a scale of spread, not confidence intervals, and give no guarantee that an entire sample path stays inside them.

Zero expected value does not make this a stationary process: its one-time probability distribution changes with $n$, as the increasing variance already proves. It does have stationary increments and independent increments, because blocks of fresh steps have the same probability distribution at any starting time and disjoint blocks use independent random variables. This separation of position from increment is one reason the random walk is such a useful first example of a stochastic process.

For a finite graph vertex set $W$ in a locally finite graph, let ${\partial }^{+}W$ be its exterior graph neighbours. The number of trifurcation vertices in percolation lying in $W$ is at most $|{\partial }^{+}W|$.

For each cluster meeting $W$, contract its connected components of a graph outside $W$ to terminal graph vertices, keeping only those adjacent to $W$. The resulting incidence graph is finite and connected. Each trifurcation graph vertex in $W$ separates at least three groups of terminals, because every infinite branch must leave the finite set $W$. Take a minimal subtree connecting all terminals. Each such trifurcation is unavoidable in that subtree and has degree of a vertex at least three. All leaves are terminals. The tree identity ${\sum }_v(\mathrm{deg}(v)-2)=-2$ bounds the number of these branching graph vertices by the number of leaves minus two, hence by the number of terminals. Different terminals, even across different clusters, can be assigned different exterior graph neighbours. Summing proves the bound. For square boxes in ${\mathbb{Z}}^2$, the boundary has order $n$ and the volume has order $n^2$.

For an infinite connected locally finite graph with a uniform degree bound, let ${\sigma }_n(v)$ count its length-$n$ self-avoiding walks from $v$. The suprema satisfy ${\sigma }_{m+n}\le {\sigma }_m{\sigma }_n$, so the Fekete lemma applied to their logarithms gives a finite connective constant. Unlike the usual rooted definition on a vertex-transitive graph, this definition explicitly takes the supremum over roots.

If independent edges of a locally finite graph receive fair orientations, exploring outward from a root tests each fresh reached-to-unreached boundary edge with success probability $1/2$. Deferred decisions give exactly the exploration law of the root cluster in bond percolation of density $1/2$. Thus the two reachable graph vertex sets have the same distribution. An infinite out-cluster contains an infinite simple directed ray by the König infinity lemma. Geometrically biased orientations do not generally give the same uniform edge-success law.