Theorem [240] The graph σ is a spanning arborescence with root r , and

arborescence A with probability proportional toα(A). Sinceα(A)contains no diagonal element pz,z of P, and each x (6=r ) is the head of a unique

edge of A, we may assume that pz,z=0 for all z∈S.

Let r _∈S. Wilson’s algorithm is easily adapted in order to sample from

6r. Letv1, v2, . . . , vn−1be an ordering of S\ {r}. 1. Letσ0= {r}.

2. Sample a Markov chain with transition matrix P beginning atvi1 with

i1 = 1, and stopped at the first time it hits σ0. The outcome is a (directed) walk W1=(u1=v1,u2, . . . ,uk,r). From W1, we obtain the loop-erased pathσ1=LE(W1), joiningv1to r and containing no loops.

3. Find the earliest vertex,vi2 say, of S not belonging toσ1, and sample a Markov chain beginning atvi2, and stopped at the first moment it hits some vertex ofσ1. Call the resulting walk W2, and loop-erase it to obtain some non-self-intersecting path LE(W2)fromvi2 toσ1. Set σ2=σ1∪LE(W2), the union ofσ1and the directed path LE(W2). 4. Iterate the above process, by loop-erasing the trajectory of a Markov

chain starting at a new vertexvij+1 ∈/ σj until it strikes the graphσj previously constructed.

5. Stop when all vertices have been visited, and setσ ₌ σN, the final

value of theσj.

2.8 Theorem [240]. The graphσis a spanning arborescence with root r , and

P(σ ₌A)_∝α(A), A_∈6r.

Since S is finite and the chain is assumed irreducible, there exists a unique stationary distributionπ₌(πs: s∈S). Suppose that the chain is reversible

with respect toπ in that

πxpx,y =πypy,x, x,y∈S.

As in Section 1.1, to each edge e ₌ [x,y_iwe may allocate the weight w(e)₌πxpx,y, noting that the edges [x,yiand [y,xihave equal weight.

Let A be a spanning arborescence with root r . Since each vertex of H other than the root is the head of a unique edge of the spanning arborescence A, we have by (2.7) that α(A)₌ Q e∈Aπe₋pe₋,e₊ Q x∈S,x6=rπx = C W(A), A_∈6r,

where C ₌Cr and

(2.9) W(A)₌Y

e∈A

w(e).

Therefore, for a given root r , the weight functionsα and W generate the same probability measure on6r.

We shall see that the UST measure on G ₌ (V,E)arises through a consideration of the random walk on G. This has transition matrix given by

px,y=    1 deg(x) if x∼y, 0 otherwise,

and stationary distribution πx=

deg(x)

2_|E_| , x∈V.

Let H ₌ (V,F)be the graph obtained from G by replacing each edge by a pair of edges with opposite orientations. Now,w(e)₌πe₋pe₋,e₊ is

independent of e_∈ F, so that W(A)is a constant function. By Theorem 2.8 and the observation following (2.9), Wilson’s algorithm generates a uniform random spanning arborescenceσ of H , with given root. When we neglect the orientations of the edges ofσ, and also the identity of the root,σ is transformed into a uniform spanning tree of G.

The remainder of this section is devoted to a proof of Theorem 2.8, and it uses the beautiful construction presented in [240]. We prepare for the proof as follows.

For each x _∈S_{\ {}r_}, we provide ourselves in advance with an infinite set of ‘moves’ from x . Let Mx(i), i ≥1, x∈ S\ {r}, be independent random

variables with laws

P(Mx(i)₌y)₌px,y, y∈S.

For each x , we organize the Mx(i)into an ordered ‘stack’. We think of

an element Mx(i)as having ‘colour’ i , where the colours indexed by i are

distinct. The root r is given an empty stack. At stages of the following construction, we shall discard elements of stacks in order of increasing colour, and we shall call the set of uppermost elements of the stacks the ‘visible moves’.

The visible moves generate a directed subgraph of H termed the ‘visible graph’. There will generally be directed cycles in the visible graph, and we shall remove such cycles one by one. Whenever we decide to remove a cycle, the corresponding visible moves are removed from the stacks, and

2.2 Wilson’s algorithm 27 a new set of moves beneath is revealed. The visible graph thus changes, and a second cycle may be removed. This process may be iterated until the earliest time, N say, at which the visible graph contains no cycle, which is to say that the visible graph is a spanning arborescenceσ with root r . If N <_∞, we terminate the procedure and ‘output’σ. The removal of a cycle is called ‘cycle popping’. It would seem that the value of N and the outputσwill depend on the order in which we decide to pop cycles, but the converse turns out to be the case.

The following lemma holds ‘pointwise’: it contains no statement involv- ing probabilities.

2.10 Lemma. The order of cycle popping is irrelevant to the outcome, in