ABSTRACT
We can generalize the random process leading to the RW by enlarging the choices we have when we reach a given vertex: before we could stop or continue. Now this last choice is enlarged to branching: the RW is allowed to branch into a number of independent RWs, a process which can be repeated. The process will in this way generate a tree-graph, i.e. a graph which contains no loops. One can draw the abstract graph in https://www.w3.org/1998/Math/MathML"> ℝ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and we will distinguish graphs which differ by orientation as shown in Fig. 4.1 (in the actual physical systems to which such trees are approximations, this turns out to the natural thing to do). We call these graphs planar branched polymers or planar trees (but we will drop the “planar” from now on). We will mainly use the notation “branched polymers” (BP), since this was the notation used when physicists meet these objects in the study of string theory, but in general, and in particular in mathematics, the “tree” notation is used. Let v denote a vertex on the abstract BP graph G and https://www.w3.org/1998/Math/MathML"> V ( G ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the set of vertices on G. We will always assume that G is a connected graph. We can assign points https://www.w3.org/1998/Math/MathML"> x ( v ) ∈ ℝ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to the vertices, and if v and v ′ are connected by a link in G we will also connect https://www.w3.org/1998/Math/MathML"> x ( v ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> x ( v ′ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in https://www.w3.org/1998/Math/MathML"> ℝ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> by a straight line. In this way G is mapped to a graph https://www.w3.org/1998/Math/MathML"> G ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in https://www.w3.org/1998/Math/MathML"> ℝ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> which we also denote a BP. We now associate an action with this https://www.w3.org/1998/Math/MathML"> ℝ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> graph: https://www.w3.org/1998/Math/MathML"> S [ G ( x ) ] = ∑ 〈 v v ′ 〉 φ ( | x ( v ) − x ( v ′ ) | ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math4_10.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
