# Euler tours and unicycles in the rotor-router model

## Abstract

A recurrent state of the rotor-routing process on a finite sink-free graph can be represented by a unicycle that is a connected spanning subgraph containing a unique directed cycle. We distinguish between short cycles of length 2 called ”dimers” and longer ones called ”contours”. Then the rotor-router walk performing an Euler tour on the graph generates a sequence of dimers and contours which exhibits both random and regular properties. Imposing initial conditions randomly chosen from the uniform distribution we calculate expected numbers of dimers and contours and correlation between them at two successive moments of time in the sequence. On the other hand, we prove that the excess of the number of contours over dimers is an invariant depending on planarity of the subgraph but not on initial conditions. In addition, we analyze the mean-square displacement of the rotor-router walker in the recurrent state.

Keywords: rotor-router model, Euler walks, uniform spanning trees, unicycles.

## I Introduction

The rotor-router walk is the latter and most frequently used name of the model introduced independently in different areas during the last two decades. The previous names ”self-directing walk” P96 () and ”Eulerian walkers” PDDK () reflected its connection with the theory of self-organized criticality BTW () and the Abelian sandpile model Dhar (). Cooper and Spencer CS () called the model ”P-machine” after Propp who proposed the rotor mechanism as the way to derandomize the internal diffusion-limited aggregation. Later on, several theorems in this direction have been proved in LP05 (); LP07 (); LP08 (). Holroyd and Propp HP () proved a closeness of expected values of many quantities for simple random and rotor-router walks. Applications of the model to multiprocessor systems can be found in RSW (). Recent works on the rotor-router walk address the questions on recurrence AngelHol (); HussSava (), escape rates FGLP () and transitivity of the rotor-routing action CCG ().

The connection between the Abelian sandpiles, Euler circuits and the rotor-router model observed in the original paper PDDK () was the subject of the rigorous mathematical survey HLMPPW (). An essential idea highlighted in the survey is the consideration of the rotor-routing action of the sandpile group on spanning trees in parallel with rotor-routing on unicycles. The rotor-router walk started from an arbitrary rotor configuration on a finite sink-free directed graph enters after a finite number of steps into an Euler circuit ( Euler tour) and remains there forever. The length of the circuit is the number of edges of the digraph. Each recurrent rotor state can be represented by a connected spanning subgraph which contains as many edges as vertices and contains a unique directed cycle CRST (); RetProb (); LoopingConst (). The dynamics of the rotor-router walk requires the location of the walker at a vertex belonging to the cycle. The pair is called unicycle (see Section II for precise definition). Thus, the walk passes the periodic sequence of unicycles.

A shortest cycle in the unicycle is the two-step path from a given vertex to one of nearest neighbors and back. We call the cycles of length 2 ”dimers” by analogy with lattice dimers covering two neighboring vertices. Longer cycles involve more than two vertices and form directed contours. The Euler tour passes sequentially unicycles containing cycles of different length. The order in which dimers and contours alternate depends on the structure of the initial unicycle. Ascribing to each step producing a contour and to a dimer, we obtain for a ”displacement” after time-steps the picture (Fig.1) resembling the symmetric random walk. Nevertheless, the process actually is neither completely symmetric nor completely random.

It is the aim of the present paper to investigate statistical properties of unicycles as they appear in course of the Euler tour.

We will see that the events ”dimer” and ”contour” correlate along the sequence and an excess of the number of dimers over contours is an invariant characterizing topology of the surface where the rotor-router walk occurs. Specifically, in the limit of large square lattice with periodic boundary conditions we find the expected number of dimers in the Euler tour and an analytical expression for the correlations dimer-dimer and contour-contour at two successive moments of time in the circuit. We consider a closed loop encircling a plane domain and prove that the rotor-router walk passed each directed edge of the domain contains the number of dimers exceeding that of contours exactly by . This property does not hold for surfaces of the non-zero genus.

In addition to statistics of unicycles, we consider the mean-square displacement of the rotor-router walker in the recurrent state and argue that it yields to the diffusion law with the diffusion coefficient depending on dynamic rules and boundary conditions.

## Ii The model

Consider a directed graph (digraph) with the vertex set and the set of directed edges without self-loops and multiple edges. If for each edge directed from to , there exists an edge directed from to , graph is bidirected. The bidirected graph can be obtained by replacing each edge of an undirected graph with a pair of directed edges, one in each direction.

A subgraph of a digraph is a digraph with vertex set and edge set being a subset of , i.e. . In this case we write . If contains no outgoing edges from a fixed vertex, that vertex is a sink. The oriented tree with sink is a digraph, which is acyclic and whose every non-sink vertex has only one outgoing edge. If the subgraph of is a tree with sink then it is called a spanning tree of with root . A connected subgraph of an oriented graph , in which every vertex has one outgoing edge, is called unicycle. The unicycle contains exactly one directed cycle.

An Euler circuit (or Euler tour) in a directed graph is a path that visits each directed edge exactly once. If such a path exists, the graph is called Eulerian digraph. A digraph is strongly connected if for any two distinct vertices , there are directed paths from to and from to . A strongly connected digraph is Eulerian if and only if for each vertex the in-degree and out-degree of are equal. In particular, the one-component bidirected graph is Eulerian. We call an Eulerian digraph with sink if it is obtained from an Eulerian digraph by deleting all the outgoing edges from one vertex. The subset of sites of connected with the sink forms an open boundary.

The rotor-router model is defined as follows. Consider an arbitrary digraph . Denote the number of outgoing edges from the vertex by . The total number of edges of is . Each vertex of is associated with an arrow, which is directed along one of the outgoing edges from . The arrow directions at the vertex are specified by an integer variable , which takes the values . The set defines the rotor configuration (the medium). Starting with an arbitrary rotor configuration one drops a chip to a vertex of chosen at random. At each time step the chip arriving at a vertex , first changes the arrow direction from to , and then moves one step along the new arrow direction from to the corresponding neighboring vertex. The chip reaching the sink leaves the system. Then, the new chip is dropped to a site of chosen at random.

In the absence of sinks the motion of the walker does not stop. The rotor configuration can be considered as a subgraph of with the set of vertices and the set of edges obtained from the arrows. The state of the system (single walker + medium) at any moment of time is given by the pair of the rotor configuration and the position of the chip . According to arguments in PDDK (), the rotor-router walk started from an arbitrary initial state passes transient states and enters into a recurrent state, continuing the motion in the limiting cycle which is the Eulerian circuit of the graph. The basic results about the rotor-router model on the Eulerian graphs can be summarized as two propositions.

Proposition 1 [HLMPPW (), Theorem 3.8] Let be a strongly connected digraph. Then is a recurrent single-chip-rotor state on if and only if it is a unicycle.

The rotor states that are not unicycles are transient. In contrast to recurrent states, they appear at the initial stage of evolution up to the moment when the system enters into the Eulerian tour.

Proposition 2 [HLMPPW (), Lemma 4.9] Let be an Eulerian digraph with edges. Let be a unicycle in . If one iterates the rotor-router operation times starting from , the chip traverses an Euler tour of , each rotor makes one full turn, and the state of the system returns to .

## Iii The unicycles on torus

Below, we specify the structure of graph as the square lattice with periodic boundary conditions (torus). Then the number of outgoing edges is for all vertices . We consider two ways of labeling of four directions of the rotor corresponding to the clockwise and cross order of routing. For the clockwise routing, we put , for the cross one . Sometimes, we will use the notation for the edge outgoing from in direction .

By Proposition 1, each recurrent state of the rotor-routing process is a unicycle on . Replacing all arrows of the rotor configuration by directed bonds, we obtain a cycle-rooted spanning tree of . It consists of a single cycle of length (i.e. directed bonds connecting vertices) and the spanning cycle-free subgraph whose edges are directed towards the cycle. If , the cycle is ”dimer”, if , the cycle is ”contour” oriented clockwise or anticlockwise.

By Proposition 2, the walker started from unicycle traverses an Euler tour which has the length in our case of torus. The questions arise how many of unicycles passed during the Euler tour contain a dimer (contour)? What is the probability that the unicycle obtained on -th step of the Euler tour contains a dimer (contour)?

Consider the recurrent state and define a random variable as

(3.1) |

We are interested in the average over all possible uniformly distributed recurrent states and over all directions .

We write the unicycle as separating off the edge and the spanning tree obtained from edges outgoing from vertices of the set . By the definition of Euler tour, all unicycles following the initial unicycle are different. If two Euler tours have a common element, they coincide.

Now, we fix a vertex and its outgoing edge . If one scans over all possible initial spanning trees , then the trees also scan over all possible configurations. So, the uniform distribution of induces the uniform distribution of . Therefore, is the probability that the edge taken uniformly with and added to the uniformly distributed spanning trees creates a dimer. Due to the translation invariance, this average does not depend on the position of the initial vertex .

To make these arguments more explicit, consider all possible unicycles

(3.2) |

for fixed vertex and arrow . First, we take , choosing the arrow at directed North. The set of unicycles can be divided into two subsets and where the first subset corresponds to spanning trees containing a selected bond incident to from above. The tree has the root in , so this bond is directed down. The selected bond and arrow form together a vertical dimer with the lower end in . In the subset , the place of the selected bond in each tree is empty, so the arrow belongs to a contour. Considering similar subsets for other directions with selected bonds of the trees incident to from right, down and left, we can write the average probability to find a dimer incident to as

(3.3) |

where summation is over all and is the total number of non-rooted spanning trees. Now, let us take the sum over all in the numerator and denominator using the uniformity of vertices of the torus. Then, the numerator will be the doubled number of edges of the spanning tree multiplied by because each edge is taken in two directions. The denominator will be . The number of edges of the torus is , the number of edges of the spanning tree . Therefore, the probability of a dimer is

(3.4) |

The probability of a contour is .

In the limit , , we obtain . In spite of this simple symmetric result, the distribution of the random value is not trivial. We will return to this question in the next section. Now consider the correlations dimer-dimer and dimer-contour at two successive moments of time in the Euler tour.

In the case of clockwise routing where the directions of each arrow alternate –––, two successive directions of an arrow at a fixed vertex always form the angle . For the cross routing (–––), the rotations – and – form the angle , whereas the rotations – and – form the angle .

The correlations we are going to determine have the following origin. Consider for example a particle arriving to vertex from above at the time step . If the arrow at is directed North in the preceding moment of time, a vertical dimer is created with the lower vertex in . If one uses the clockwise dynamics, the next step is the rotation of the arrow at to East. Assume that there is an arrow at time directed to from right to left. Then, the horizontal dimer is created at the time step with the left vertex in . The probability to get two dimers at moments and is the correlation .

The arguments used for the derivation and show that the correlations , , and at two successive time-steps can be related with the probability to find two adjacent edges of the square lattice occupied (or not occupied) by bonds of the spanning tree .

Specifically in the considered example, we must enumerate unicycles with the vertical dimer having the lower end in . The spanning tree of the unicycle has the root and two fixed bonds, directed to from above and directed to from the right. The presence of the root in implies that all bonds of the tree are globally oriented towards the vertex .

The enumeration of spanning trees obeying the above conditions can be performed in three steps. First, we consider non-oriented spanning tree with the selected non-oriented bonds and on the places of and . Second, we put the root in , giving the necessary orientation to bonds and and supplying other bonds with the global orientation towards . Third, we use the Kirhhoff theorem according to which the number of spanning trees does not depend on the location of the root. This allows us to shift the root to infinity and restore the translation invariance in the limit and .

The alternative way of calculations would consist in fixing the oriented bonds and and the location of the root in . However, the Kirhhoff theorem does not allow the translation of the root in the presence of oriented bonds. Then, the lack of translation invariance makes all calculations much more difficult.

We fix a vertex and its two neighbors on the square lattice and . Then and are adjacent edges.

Define the probabilities

(3.5) | |||||

(3.6) | |||||

(3.7) | |||||

(3.8) |

Obviously, and due to symmetry.

The calculation of probabilities of fixed spanning tree configurations is a standard procedure, which uses the Green functions and so called defect matrices (see e.g. MajDhar (); Pri94 (); PirouxRu (); P12 (); PogPri ()). In our case, it gives

(3.9) | |||||

(3.10) | |||||

(3.11) |

where the matrices , are

(3.12) |

and the defect matrices , and are

(3.13) |

Defect matrices define the locations of bonds and which form angles or . In the first case we add index to the notations of probabilities, and index for the second case. Using the explicit values for the Green functions given in Appendix, we obtain in the limit and

(3.14) | |||||

(3.15) |

for the case (a), and

(3.16) | |||||

(3.17) |

for the case (b).

Then, for the correlations dimer-dimer and dimer-contour at two successive moments of time in the Euler tour we have

(3.18) | |||||

(3.19) |

in the case of the clockwise routing, and

(3.20) | |||||

in the case of cross routing.

## Iv The balance between dimers and contours

Consider a part of the Euler tour as a sequence of unicycles with the first element and the last element . The whole Euler tour in these notations is and the last unicycle is not included into the sequence. We define a random value as

(4.1) |

If the cycle in unicycle is a contour oriented clockwise, we denote by the unicycle which differs from only by the counter-clockwise orientation of the contour. The following proposition for the clockwise routing has been announced in PPS () and formulated in HLMPPW () as Corollary :

Let be a bidirected planar graph and let be a unicycle with the cycle oriented clockwise. After the rotor-router operation is iterated some number of times, each rotor internal to has performed a full rotation, each rotor external to has not moved, and each rotor on has performed a partial rotation so that the cycle is counter-clockwise .

Below, we prove that for any and if the subgraph surrounded by is planar and the walker moves according to the clockwise routing. It is important to note, that the clockwise routing is crucial both for the Corollary and the identity .

Using the method of induction, we start with the case of minimal when the contour is an elementary square of area 1. In this case, the walker makes four steps: starts with the clockwise contour , produces sequentially three dimers and ends by counter-clockwise contour . Denoting this sequence as we see that .

Consider now unicycle containing a clockwise contour of area . We assume that for all and prove that . Due to the symmetry of the square lattice, we can fix without loss of generality the edge with at the left side of the contour . Let be coordinates of the vertex . We consider four stages of the transformation of unicycles from to .

Stage 1. The chip moves from to along the edge . If the obtained cycle is dimer, Stage 1 is completed. Otherwise, the cycle is a contour of area . By Corollary, the unicycle with contour transforms after some number of steps into the unicycle with and by the assumption, . The contour contains the edge .

Stage 2. The chip moves from to along the edge . If the obtained cycle is dimer, Stage 2 is completed. Otherwise, the cycle is a contour of area . By Corollary, the unicycle with contour transforms into one with and by the assumption, . The contour contains the edge .

Stage 3. The chip moves from to along the edge . If the obtained cycle is dimer, Stage 3 is completed. Otherwise, the cycle is a contour of area . Again, the unicycle with contour transforms into one with and . The contour contains the edge .

Stage 4. The chip moves from to along the edge and produce the original unicycle but with the opposite orientation of the contour.

The description of evolution of unicycles from to shows that the number of dimers exceed that of contours by 1 during every stages 1,2,3 independently of whether the first or the second scenario of evolution is realized at each stage. Taking into account that the cycles of the first unicycle and the last one are contours, we obtain

(4.2) |

Remark. According to Corollary the sum .

The result (4.2) proven for the plane domain is in a drastic contrast with the rotor-router walk on surfaces of the non-zero genus. In Fig.2 the function is shown for the whole Euler tour on the torus for clockwise routing. We see that takes different values depending on and all these values are non-negative. The average is known from (3.4). Indeed, for the Euler tour of length , we have the average number of dimers and the average number of contours . Therefore =4 for any and .

To consider a more general situation, we fix a unicycle with a clockwise contour which cuts out a plane domain from the torus. According to Corollary, the contour in will be converted into the counter-clockwise contour in after some number of steps of the Euler tour started from . The contour is counter-clockwise with respect to and clockwise with respect to the complement domain of genus 1. Now, we separate the whole Euler tour into two parts: from to and from to . From (4.2) we have . Therefore, to provide non-negativity of , we should admit . A reason for the excess of contours over dimers is the existence of many additional loops on the surface of non-zero genus, in particular non-contractible loops on the torus. We are not able to prove an exact inequality for , so we formulate it as a conjecture:

Conjecture Let be a contour clockwise with respect to the surface of genus 1 and is unicycle containing . Then, for the sequence of unicycles in the Euler tour, the difference .

The conjectured inequality as well as (4.2) get broken in the case of cross routing. Fig.3 shows for the Euler tour with the cross routing rules.

Instead of the strictly asymmetric distribution in Fig.2, we have a Gaussian-like distribution with the width corresponding to the diffusion law. To check the Gaussian nature of the distribution, we calculated moments , and estimated skewness and excess kurtosis. For the lattice size with statistics of samples, we obtained and . Nevertheless, the exact normality of the distribution in the limit and remains an unproved conjecture.

The average coincides with that for the clockwise routing because the probability (3.4) does not depend on the order of routing.

## V The diffusion of the walker

Given the Euler tour of length on the torus , we can find the mean-square displacement after steps, where and are coordinates of the walker at time . Fig.4 shows for two periods of the Euler tour with the clockwise routing. The interpolation of the function in the interval gives the linear dependence .

The obtained linear law is not surprising. The time dependence of mean square displacement cannot be slower than , where is a constant. Indeed, by the definition of Euler tour, each vertex of the torus cannot be visited more than 4 times. Therefore, the walker cannot stay in an area of radius longer than time steps.

On the other hand, cannot be faster than where is another constant. It follows from the Corollary that the walk is ”loop-fiiling”, i.e. the interior of a loop of radius is visited densely, so that each rotor inside the loop makes a full rotation before the walker leaves the loop. Therefore, an advance of the walker at the distance of order takes steps.

The exact value of the diffusion constant is unknown. The computer simulations show that it depends on the order of routing and we can estimate it as:

(5.1) | |||||

(5.2) |

## Acknowledgments

This work was supported by the RFBR grants No. 12-01-00242a, 12-02-91333a, the Heisenberg-Landau program, the DFG grant RI 317/16-1 and State Committee of Science MES RA, in frame of the research project No. SCS 13-1B170.

## Vi Appendix

The translation invariant Green function for the infinite square lattice is spitz ()

(6.1) |

with an irrelevant infinite constant . The finite term is given explicitly by

(6.2) |

and obeys the symmetry relations:

(6.3) |

After the integration over , it can be expressed in a more convenient form,

(6.4) |

where , .

Below, we give for several values which are used in the text

(6.5) |

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