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the rotor mechanism , firstly proposed in the theory of self - organized criticality @xcite under name `` eulerian walk '' @xcite , was rediscovered independently as a tool for a derandomization of the random walk @xcite . the subsequent studies were concerned with collective properties of the medium `` organized '' by the walk and with statistical properties of the walk itself @xcite . the dynamics of the rotor - router walk can be described as follows . consider a square lattice with arrows attached to the lattice sites . arrows attached to the lattice sites are directed toward one of their neighbors on the lattice . a particle called usually _ chip _ , performs a walk jumping from a site to a neighboring site . arriving to a given site , the chip changes direction of the arrow at that site in a prescribed order and moves toward the neighbor pointed by new position of the arrow . thus , given an initial orientation of arrows on the whole lattice , the rotor - router walk is deterministic . the walk started from uniformly distributed random initial configurations can be called uniform rotor walk . three steps of the rotor walk on the square lattice are shown in fig.[steps ] .        if the lattice is finite , the walk starting from an arbitrary site settles into an eulerian circuit where each edge of the lattice is visited exactly once in each direction @xcite . when the walker is in the eulerian circuit , configurations of rotors @xmath4 associated to each site are recurrent . a graphic representation of the recurrent configuration is _ unicycle _ which is a specific state where the arrows form a spanning set of directed edges containing a unique directed cycle which the chip belongs to @xcite . if the position of the chip on the cycle is @xmath5 , we denote the unicycle as @xmath6 .    along with the unicycle , we can define the _ multicycle _ @xcite as a graph containing exactly @xmath7 cycles together with @xmath7 chips at vertices @xmath8 belonging to the cycles . for multicycles , we use the notation @xmath9 .    for the infinite lattice , both questions on trajectories of the walker and on the configurations of arrows become more complicated . a basic problem here is to find the range of the rotor walk , i.e. the number of distinct sites visited in @xmath10 steps and , given the lattice symmetry and the rotor mechanism , to find a shape of the area visited by the walker . one conjecture and one theorem shed light on this problem . kapri and dhar @xcite conjectured that the set of sites visited by the clockwise uniform rotor walk in @xmath10 steps on the infinite square lattice is asymptotically a disk of average radius @xmath11 where @xmath12 is a constant . florescu , levine and peres @xcite proved that for an infinite @xmath13-dimensional lattice , regardless of a rotor mechanism or an initial rotor configuration , the rotor walk in @xmath10 steps visits at least on the order of @xmath14 distinct sites . monte carlo simulations in @xcite showed that the average number of visits of a site inside the disk is a linear decreasing function of its distance from the origin . the authors of @xcite give the following explanation of this characteristic behavior . after a moment when two sites at different distances from the origin are visited by the rotor walk , both sites are visited equally often because of the local euler - like organization of arrows . then , the difference between the numbers of visits of these sites remains bounded for an arbitrary number of subsequent steps . the key point in this explanation is the local eulerian organization which is proven rigorously only for finite graphs as a part of the total organization . for the infinite lattice , any bounded domain tends to the entirely organized state only asymptotically being repeatedly visited by the rotor walk . a question , however , is in the periodicity of returns . the mean number of returns and the mean - square displacement should be in a definite proportion to provide the sub - diffusive behavior of the rotor walk . so , it is desirable to find in the system of rotors some structure which provides sufficiently often returns of the walker to the origin and , as a consequence , to any previously visited site . the construction of such a structure is the main goal of the present paper .    in the recent work @xcite , we have considered the motion of the clockwise rotor - router walk inside closed contours emerged in random rotor configurations on the infinite square lattice . we proved a property called the _ weak reversibility _ : even though the configuration of rotors inside the contour is random , the rotor - router walk inside the contour demonstrates some regularity , namely , the chip entering the clockwise contour @xmath15 in a vertex @xmath16 leaves the contour at the same vertex @xmath17 , and then the clockwise orientation of rotors on @xmath15 becomes anti - clockwise . we referred to the sites where rotors complete clockwise contours as _ labels _ , and noticed that the sequence of labels forms a spiral structure . after averaging over initial random configurations of rotors , the sequence approaches asymptotically the _ archimedean _ spiral . however , the spiral structure as such does not explain the obligatory periodic visits of the origin by the rotor walk . in section iii , we consider particular labels called _ nodes_. the set of nodes being a subset of that of labels has also the spiral structure . the difference between labels and nodes lies in the disposition of contours corresponding to them . in the case of labels , a contour completed at given site is not necessarily adjacent to the contour associated with the previous label . in case of nodes , each new contour associated with a node either has common sites with that corresponding to the previous node , or contains this contour inside .    in section iv , we analyze the structure of contours associated with nodes . according to the week reversibility , each contour after visiting its interior becomes anti - clockwise and left by the chip . at the moment of exit from the node located at vertex @xmath17 , there is a directed path formed by arrows and connecting one of neighboring sites of @xmath17 with the previous node . a collection of paths obtained at the moment @xmath10 is a tree rooted at the current location of the chip . the tree structure together with spiral - like motion of the chip provides the obligatory visits to the origin for each turn of the spiral . depending on the location of the origin with respect to the first clockwise contour , the chip returns to the vicinity of origin by different ways . once the spiral structure is formed , the number of visits to the origin is 4 for each rotation around the origin . then , the total average number of visits @xmath0 for @xmath3 rotations performed by the chip starting from the uniform random initial configurations of arrows is @xmath2 , when @xmath18 . since the archimedean spiral has a constant interval between coils , we obtain the linear dependence between the radius of the spiral and the number of returns to the origin .    in the separate section v , we analyze the convergence of the set of labels ( nodes ) to the archimedean spiral . as it was noticed in @xcite , this convergence is extremely slow . the existence of the limit for the averaged ratio of radius @xmath19 to angle @xmath20 which is a constant @xmath21 for the purely archimedean case , is not proven yet . a prospective value of @xmath21 can be obtained from the scaling law for the average number of visits conjectured by kapri and dhar @xcite . extensive simulations show that the deviation from the constant @xmath21 remains considerable for very large number of nodes @xmath22 and huge number of steps @xmath23 . we discuss a possible reason for this deviation . we consider the infinite square lattice , and fix the clockwise rotor mechanism at each site . in the initial rotor state , the arrows at each lattice site are directed randomly to one of four directions with equal probabilities , and the chip is in the origin . at each step of discrete time , the chip arriving at a site rotates the arrow at that site 90 degrees clockwise , and moves to the neighboring site pointed by the new position of arrows . the motion of the chip is determined by the current rotor state . given a rotor state , we say that a group of arrows forms a directed path if the arrows are attached to sites @xmath24 such that @xmath25 and @xmath26 are neighbors , and the arrow at @xmath25 is directed toward @xmath26 for @xmath27 . the directed path of arrows becomes a cycle if @xmath28 . a shortest possible cycle consists of two adjacent sites @xmath29 , @xmath30 , which are connected by a pair of edges from @xmath29 to @xmath30 and back . we call such cycles _ dimers _ by analogy with lattice dimers covering two neighboring sites . a cycle formed by more than two edges is called _ contour_.    the configuration of arrows inside a contour is either free of cycles or contains a number of cycles . in the first case , the arrows inside the contour form a spanning forest rooted at the contour . in the second case , the arrows form a spanning forest where each tree is rooted either at the external contour , or at one of the internal cycles . correspondingly , two theorems describe the behavior of the chip approaching contours . _ theorem 1 _ on reversibility ( @xcite and @xcite , corollary 4.11 ) . let @xmath31 be a planar graph containing a unicycle @xmath6 with the contour @xmath15 oriented clockwise and @xmath32 . after the rotor - router walk makes some number of steps , each rotor internal to @xmath15 has performed a full rotation , each rotor external to @xmath15 has not moved , each rotor on @xmath15 has performed a partial rotation so that @xmath15 is now oriented anti - clockwise and the chip has returned to @xmath5 .    to describe the motion of the chip in the second case , we consider a more general situation , where several chips are involved into the evolution of arrows inside the contour .    _ theorem 2 _ , @xcite . let @xmath31 be a connected bidirected planar graph and @xmath9 be a multicycle with the external contour @xmath33 oriented clockwise together with @xmath34 internal cycles @xmath35 oriented anti - clockwise . the rotor - router operation is sequentially applied to the chip at @xmath36 until the moment @xmath37 when the chip returns to @xmath38 , and the rotor at @xmath38 is made oriented anticlockwise . then , the same is applied to chips at @xmath39 until the moments @xmath40 when chips starting from @xmath41 return to @xmath42 and the rotors at @xmath42 are made oriented clockwise . then , all rotors on @xmath33 are becoming oriented anticlockwise , all rotors on @xmath35 become oriented clockwise , and all vertices internal to @xmath33 and external to @xmath35 perform a full rotation . the description of motion of the single chip inside a contour is given by a reduced version of theorem 2 :    _ theorem 3 _ on weak reversibility . let @xmath31 be a planar graph containing the external contour @xmath33 oriented clockwise and some number of internal cycles inside @xmath33 . the rotor - router walk starting at site @xmath17 , @xmath43 moves until the moment when the chip returns to @xmath17 . as a result , all rotors on @xmath33 become oriented anti - clockwise , and the chip leaves contour @xmath33 at the next time step . the rotors internal to @xmath15 perform either a full rotation or a partial rotation or do not move at all . the theorem 2 allows us to specify internal sites of contour @xmath33 which perform the full rotation . _ corollary . _ let @xmath44 be a set of sites inside the clockwise contour @xmath33 which belong to the forest rooted at @xmath33 . assume that there is the site @xmath45 and @xmath46 . the chip starting at site @xmath43 moves until the moment when the chip returns to @xmath17 . then , the rotor at @xmath47 performs the full rotation . _ let @xmath48 be the total number of steps in the process described in theorem 2 . the process can be divided into two stages , i and ii : stage i for steps @xmath49 and stage ii for steps @xmath50 . consider the site @xmath51 belonging to the forest @xmath44 rooted at @xmath33 . according to theorem 2 , the rotor at @xmath47 performs the full rotation to the moment @xmath52 . the number of rotations is @xmath53 . assume that a part of rotations is performed during stage i and the rest of them during stage ii . since stage ii follows stage i , a sequence of arrows resulting after @xmath52 steps is directed from @xmath47 to one of sites @xmath54 . but @xmath47 belongs to a tree rooted at @xmath33 by the condition . therefore , all @xmath53 rotations are performed during stage i , which coincides with the process described in theorem 3 . below , we apply the theorems 1,3 and corollary to investigate the uniform rotor walk on the infinite square lattice . the rotor walk during the time evolution creates contours of arrows @xmath55 sequentially . in @xcite , we considered the set of sites @xmath56 where each contour becomes closed at time steps @xmath57 , and called these sites labels . according to theorems 1,3 , the rotor walk leaves each contour @xmath58 at step @xmath59 at the same site @xmath60 . we skip details of evolution in the time intervals between @xmath61 and @xmath62 and ignore possible new contours appearing inside @xmath58 during these intervals . the only fact of the evolution between @xmath61 and @xmath62 we take into account is reversing the clockwise orientation of the contour @xmath58 . it was found in @xcite that the labels @xmath56 are not simply situated in the cluster of visited sites but form a spiral structure . an example of the spiral of labels is shown in fig.[fig2](a ) . whereas each particular spiral has an irregular form , their average over uniform random initial rotor states tends to the spiral obeying the archimedean property @xmath63 in planar coordinates @xmath64 , with constant @xmath5 and @xmath21 . a convergence of spirals to the archimedean law ( [ archimed ] ) for a large number of steps is discussed in section v.    despite the surprising property , the spiral of labels does not say anything about periodicity of the chip returns to the vicinity of the origin needed for organization of the cluster of visited sites . to answer this question , we consider the sequence of contours @xmath55 corresponding to labels in more detail . let @xmath58 and @xmath65 be two successive contours in the sequence @xmath55 . there are three possibilities for the disposition of @xmath65 with respect to @xmath66 : a ) the set of sites @xmath67 where arrows of @xmath65 are attached has no common sites with @xmath68 and contour @xmath65 is outside @xmath66 ; b)the set @xmath67 has no common sites with @xmath68 and contour @xmath66 is inside @xmath65 ; c ) the set @xmath67 has at least one common site with @xmath68 . to provide for the condition b ) , the contour @xmath65 should contain inside at the moment @xmath69 all sites visited at moments @xmath70 . otherwise , there are lattice sites outside @xmath65 which do not connected with @xmath58 at the moment @xmath61 by any path of arrows , what is impossible for a single walk . when the cluster of visited sites grows , the probability of a contour enveloping the cluster of previously visited sites dramatically decreases and we can exclude the case b ) from the consideration . then , we select from the set of labels @xmath56 a subset of labels @xmath71 whose contours obey criterion c ) and assume that @xmath72 coincides with @xmath29 . we call the selected labels _ nodes_. fig.[fig2](b ) shows the spiral consisting of nodes selected from the labels of fig.[fig2](a ) . let @xmath73 be the sequence of nodes generated by the rotor walk . by the construction , at the moment of exit from the node located at vertex @xmath74 , there is a directed path formed by arrows and connecting one of neighboring sites of @xmath74 with the previous node @xmath75 . thus , the system of directed paths and remaining parts of contours associated with nodes forms a connected network . in what follows , we will see that the obtained network is a tree constructed from topologically uniform elements . the chip moving from a fixed site @xmath76 to site @xmath77 traces a path of arrows directed from @xmath76 to @xmath77 . fig.[labstar](a ) illustrates the path traced by the chip moving between two successive nodes @xmath78 and @xmath79 selected from the sequence of nodes @xmath73 . the first node is at the site @xmath78 marked by 1 . the directed path between sites 6 and 1 is a part of the contour @xmath80 corresponding to the first node . the chip creates a clockwise contour at site 2 , reverses it to the anti - clockwise one and leaves it at the same site 2 . assume , that this contour has no common sites with line @xmath81 . if it also has no intersections with either of previous contours , the site 2 is a label but not a node . the situation is repeated at site 3 and the chip continues motion creating an arbitrary number of labels until reaching site 5 . in general , site 5 does not belong to line @xmath81 because of a possible sequence of arrows directed from site 5 to site 6 existing before the chip could reach it . as a result , the clockwise contour appears @xmath82 at site 5 which is a node @xmath74 according to criterium c ) . after reversing @xmath83 to the anti - clockwise contour @xmath84 , the chip leaves @xmath85 at site 5 . the parts of reversed contours corresponding to the labels 2 and 3 and possible others are just branches attached to the path from 4 to 5 which do not affect connectivity of this path and therefore can be ignored . the resulting path shown in fig.[labstar](b ) is a building block of the tree we are going to construct . the tree is constructed by consecutive adding the building blocks , one by one for each new node . in fig.[tree ] we show schematically how the tree grows if one neglects a possible difference in sizes of the blocks and takes into account their topological structure only . consider the first clockwise contour containing the origin . in fig.[tree ] , this contour is formed by the path starting at site @xmath86 and reaching site 2 . due to the possible directed sequence from site 2 to 3 , the contour is closed at site 2 which is the first node . at site 2 , the chip leaves the contour after reversing its direction and reaches site 5 which is connected with site 6 by the directed sequence of arrows , if exists ( otherwise site 5 coincides with 6 ) . then the clockwise contour 2,4,5,6,3,2 appears with the next node at site 5 . the interior of this contour covers the sector of space bounded by two radial branches 3,2,4 and 3,6,5 . reversing this contour to the anti - clockwise one , the chip continues motion from node 5 to node 8 and so on . by the construction of the building blocks , the resulting graph @xmath87 is a tree . at each moment of time , all directed paths of arrows constituting the tree are oriented towards the current position of the chip . consider the chip at the last node situated at site 20 . its further path can trace a directed path of arrows to any reachable site of the tree @xmath87 . assume this site is on the interval between sites 10 and 8 . then the next clockwise contour will contain the sites 8,9,6,3,2,21,20 and cover the sector bounded by radial branches 3,2,21,20 and 3,6,9,8 . the continuation of the spiral up to full rotation creates subsequent contours covering one by one _ all _ sectors between branches of the tree , including the sector bounded by line @xmath88 and containing the origin . if the initial rotor configuration contains one , two or three sequences of arrows flowing into the origin , the structure of the tree @xmath87 admits two , three or four sectors having the origin on the boundary between sectors .    the described scenario being empirical is nevertheless typical for any random initial configuration of rotors . the obtained tree is stable after each rotation of the spiral . indeed , by theorem 2 , the chip visiting a contour changes its orientation but not its form . branches of the tree situated inside contours consist of sites which perform the full rotation according to the corollary in section ii and therefore remain stable as well . the arrows inside the contours not belonging to these branches can not create new cycles due to the rotor mechanism and can only add new branches to the existing tree . the main conclusion we can draw from the existence of the tree structure and the spiral ordering of nodes is that every turn of the spiral generates necessarily either a contour containing the origin inside it or generates contours containing the origin on their boundaries . now we are ready to answer the question about the number of visits to the origin for each rotation . consider the first clockwise contour @xmath89 containing the origin inside . by the construction of the tree @xmath87 , the origin @xmath86 belongs to the forest rooted at sites of @xmath89 . it follows from the corollary , that the arrow at the origin performs the full rotation when @xmath89 is reversed . as the tree @xmath87 grows , the subsequent contours containing the origin become larger and the chip visiting them not necessarily visits all sites of their interiors . however , the forest arisen in the first contour remains rooted at all subsequent contours containing the origin . then by the corollary , the rotor at the origin performs the full rotation each time when such a contour appears . since it happens for each turn of the spiral , the number of visits of the origin @xmath90 depends of the number of turns @xmath91 as @xmath92 where @xmath93 is due to possible visits to the origin before the moment when the first loop of the spiral is formed . the result of simulation for a single spiral is shown in fig.[s1 ] . vs number of rotations @xmath91.,width=264 ]    if the initial configuration contains one or more sequences of arrows directed to @xmath86,the origin can belong to the boundary of adjacent sectors of the tree . since all sectors are covered by contours appearing during one rotation of the spiral , the total number of visits to the origin remains 4 . for instance , let @xmath94 and @xmath95 be two adjacent contours and the origin 0 is at the boundary between them . reversing contour @xmath94 at some stage of rotation , the chip rotates the arrow at 0 by angle @xmath96 or @xmath97 depending on the form of the boundary . contour @xmath95 is also reversed during the same turn with the arrow rotation by the additional angles @xmath98 or @xmath99 giving the full rotation per one turn . the emergence of a spiral structure in the clockwise rotor - router walk comes from a rather simple reason . indeed , consider a label @xmath100 situated near the boundary of the cluster of visited sites . a preferable position for the next label @xmath101 is on the right side from the branch @xmath102 to provide the clockwise orientation of the contour @xmath103 if it has common edges with @xmath102 . then , the preferable direction of successive positions of labels @xmath104 is clockwise with respect to the origin of the cluster . since the size of the cluster grows with time , the positions @xmath104 form a spiral - like structure . the set of nodes , being a subset of labels has the spiral form as well . moreover , the condition for contours corresponding to two successive nodes to be adjacent imposes an additional restriction on the positions of nodes . the comparison of typical spirals in fig.[fig2](a ) and ( b ) shows that this restriction makes the spiral of nodes more regular than that of labels . introducing the polar coordinates , we denote by @xmath105 the distance from the origin and by @xmath106 the winding angle of @xmath7-th node . then , we say that the spiral of nodes is asymptotically archimedean in average if @xmath107 where the average is taken over the uniformly distributed states of the spiral . a numerical verification of the archimedean property for labels in @xcite showed a very slow converge to the asymptotic law ( [ definition ] ) . here , we use the more pronounced spiral structure of nodes to determine the asymptotic behavior of @xmath108 with greater accuracy . first , we compare the conjectured archimedean property ( [ definition ] ) with the conjecture by kapri and dhar @xcite , who supposed that the average number of visits @xmath109 to the site separated from the origin by distance @xmath110 for @xmath111 steps satisfies the scaling form @xmath112 where @xmath113 is the scaling function @xmath114 , @xmath115 . the data obtained in @xcite for @xmath116 and @xmath117 collapse with @xmath118 and @xmath119 . then , the limiting radius of the circle of visited sites depends on the number of visits to the origin @xmath90 as @xmath120    the spiral of nodes , by the construction , is contained inside the circle of visited sites . according to the archimedean law eq.([archimed ] ) with @xmath121 , for the large number of rotations @xmath91 , the spiral curve tends to the circle of radius @xmath122 . then , due to eq.([number ] ) , we obtain again the linear dependence of the radius on the number of visits to the origin @xmath123 thus , the conjectured archimedean property is consistent with the linear scaling law eq.([scaling ] ) . in the scaling limit , we can expect that @xmath124 . then , the constant @xmath21 in eq.([definition ] ) is related to the constants of the scaling law as @xmath125 . taking @xmath118 and @xmath119 , we obtain @xmath126 . to study the asymptotical behavior of the spiral , we extended our simulations to number of time steps @xmath127 and number of samples @xmath128 . the numbers of nodes and spiral rotations we use in our analysis are shown in fig.[fig5 ] . the result of averaging of the ratio @xmath129 over uniformly distributed initial conditions as a function of the node number is shown in fig.[fig4 ] . the horizontal line in fig.[fig4 ] corresponds to @xmath126 obtained from the scaling conjecture @xcite . we can see that the deviation from the constant @xmath21 remains considerable up to very large node numbers . vs number of node @xmath7.,width=415 ]    despite the apparent decrease of the slope of the curve in fig.[fig4 ] , we can not guarantee the coincidence of the limiting value of @xmath130 with the scaling value @xmath126 and even the existence of the limit as such . the monte - carlo simulations show that the ratio of radius to angle grows with @xmath7 not faster than @xmath131 with @xmath132 . the logarithmic deviation obtained for finite @xmath7 suggests to try an asymptotic expansion in powers of @xmath133 . the obtained data yield the following lower bound : @xmath134    a reason for so slow convergence is a geometrical non - equivalence of averaging spirals . indeed , if the random spirals differ one from another only by the distances between coils , the averaging over large number of samples would give a well defined mean distance even for a relatively broad distance distribution . instead , we observe some number of meanders in the spiral structure at different regions of the spiral . this leads to an effective broadening of the intervals between coils , but since the meanders are rare events , the determination of parameters of the averaged spiral needs an enormously large statistics . vbp thanks the rfbr for support by grant 16 - 02 - 00252 . vsp thanks the jinr program `` ter - antonyan - smorodinsky '' .
we study the rotor - router walk on the infinite square lattice with the outgoing edges at each lattice site ordered clockwise . in the previous paper [ j.phys.a : math . 
 theor . 
 48 , 285203 ( 2015 ) ] , we have considered the loops created by rotors and labeled sites where the loops become closed . 
 the sequence of labels in the rotor - router walk was conjectured to form a spiral structure obeying asymptotically an archimedean property . in the present paper , 
 we select a subset of labels called `` nodes '' and consider spirals formed by nodes . 
 the new spirals are directly related to tree - like structures which represent the evolution of the cluster of vertices visited by the walk . 
 we show that the average number of visits to the origin @xmath0 by the moment @xmath1 is @xmath2 where @xmath3 is the average number of rotations of the spiral . 
 _ keywords _ : rotor - router walk , archimedean spiral , sub - diffusion .