Critical point for infinite cycles in a random loop model on trees
aa r X i v : . [ m a t h . P R ] M a y CRITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES
ALAN HAMMOND AND MILIND HEGDEA bstract . We study a spatial model of random permutations on trees with a time parameter T > T slightly above aputatively identified critical value but left open behaviour at arbitrarily high values of T . We showthe existence of infinite cycles for all T greater than a constant, thus classifying behaviour for allvalues of T and establishing the existence of a sharp phase transition. Numerical studies [BBBU15]of the model on Z d have shown behaviour with strong similarities to what is proven for trees.
1. I ntroduction
Consider a collection of points scattered independently in a large three-dimensional torus sothat any unit-volume region contains a unit order of points. For T > T Brownian trajectories in the torus, with a short-range repulsive force continually actingbetween any pair of points. The system is conditioned on the collective return at time T of theparticles to their starting locations. A random permutation is obtained by following the trajec-tory for time T of any given particle from its initial location. This mathematical spatial randompermutation model is physically significant, as first recognised by Richard Feynman in [Fey53]: athigher values of time T , large cycles may be expected to form in the random permutation, withthe reciprocal values T − corresponding to lower temperatures at which gases such as heliumform special states such as Bose-Einstein condensates or superfluids.A simple mathematical model of spatial random permutations is the random stirring pro-cess, sometimes known as the random interchange model. The model was introduced by Har-ris [Har72]. It associates to a given graph G = ( V , E ) a stochastic process (cid:0) σ t : t ∈ [ ∞ ) (cid:1) whichtakes values in the space of permutations of the vertex set V . Each edge e ∈ E is independentlyequipped with a Poisson process of rate one. We set σ to be the identity permutation. If thePoisson process on an edge e = ( v , w ) rings at time t , we right-compose σ t with the transposition ( v , w ) , i.e., we swap v and w in the permutation process, in a right-continuous manner. Themodel is easily seen to be well defined on regular infinite graphs, or indeed if the maximumdegree of the graph is finite. The formation of large cycles in σ t is a main topic of inquiry. D epartment of M athematics , UC B erkeley , 899 E vans H all & 741 E vans H all E-mail address : [email protected], [email protected] .The first author is supported by NSF grant DMS-1512908. B ´alint T ´oth [T´93] showed that the quantum Heisenberg ferromagnet has long range order andspontaneous magnetisation in a phase that corresponds to the appearance of macroscopic cyclesin a variant of the random stirring model in which permutations are reweighted by a factor oftwo for each cycle.T ´oth conjectured in the 1990s that the random stirring process on transient graphs (such as Z d for d ≥
3) exhibit a critical point above which infinite cycles almost surely appear and belowwhich they do not.When d is high, the model on regular trees of degree d may be expected to be similar to themodel on Z d . Omer Angel [Ang03] showed that on regular trees with degree at least five, thereexists a certain bounded interval of times where σ t a.s. has an infinite cycle. The existence of acritical value for infinite cycles was proved in [Ham13, Ham15].Aizenman and Nachtergaele in [AN94] introduced a representation of the quantum Heisen-berg antiferromagnet via a variant of the random stirring process in which a certain time reversaloccurs when particles are transposed due to the ringing of Poisson clocks on the associated edge.Ueltschi introduced in [Uel13] a hybrid model in which ferromagnetic and antiferromagneticeffects are both present, and with Bj ¨ornberg in [BU16] analysed the new model when the under-lying graph is a high degree tree; comparison with numerical evidence shows that the model isa very good surrogate for its Z d counterpart. The present article develops their work by provinga result that was very strongly suggested by their results: that the hybrid model has a criticalpoint for the formation of infinite cycles.1.1. The Cyclic Time Random Walk and its Modification.
In order to explain Bj ¨ornberg andUeltschi’s results and how our paper develops them, it is useful to begin by recalling a randomprocess which is a very close cousin of the random stirring process. This is the Cyclic-TimeRandom Meander (CyTRM); it is a slight variant of the cyclic-time random walk considered byAngel in [Ang03] and has been used in Angel and Hammond’s analyses of the random stirringprocess on trees.Recall that, when the random stirring process is specified, a graph G = ( V , E ) is given. TheCyTRM is defined by fixing a parameter T ∈ ( ∞ ) , and associating to each edge of G anindependent Poisson process of rate one on [ T ) . We may picture the graph’s vertices as pointsin the plane, with a vertical pole rising to height T above each of them. A horizontal bridge isplaced between the poles rising from vertices v and w at any height at which the Poisson processfor the edge ( v , w ) rings; in this case we say the edge ( v , w ) supports a bridge; see Figure 1.The CyTRM is a right-continuous random process X mapping [ ∞ ) to V × [ T ) and may bedepicted as a point moving in the union of the poles. If initially X ( ) = ( v , 0 ) for some v ∈ V , X rises vertically at unit speed on the pole at v . If it encounters a bridge’s intersection withthis pole, the process instantaneously jumps across the bridge, and then continues its unit speedascent on the newly encountered pole. When, at time T , the point reaches the top of a pole, RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 3 T φ F igure
1. An illustration of the root φ and two offspring. The vertical lines arethe poles, the dashed lines are the edges in the underlying graph, and the solidslanted lines are the bridges supported by the edges underneath at the heightsdictated by the independent rate one Poisson processes associated with each edge.it immediately jumps to the base of the same pole. Vertical ascent then continues, so that theprocess’ height at times t ≥ t mod T .The random stirring process σ T at parameter T ∈ ( ∞ ) is formed from cyclic-time randommeander X as the permutation on vertices induced by the evolution of X during the interval [ T ] . Formally, when X ( ) = ( v , 0 ) for given v ∈ V , we have that X ( T ) = (cid:0) σ T ( v ) , 0 (cid:1) . Thepresence of an infinite cycle containing v ∈ V in σ T is characterised by the absence of return toits starting point by cyclic-time random meander with X ( ) = ( v , 0 ) ; see [Ang03] for details.The random loop model introduced by Bj ¨ornberg and Ueltschi in [BU16] is a generalisationof cyclic-time random meander. Given a parameter u ∈ [
0, 1 ] , and the structure that specifiesthe meander, independently assign to each bridge a Bernoulli random variable of parameter u .When the random variable equals one, the bridge is replaced by a cross ; and by a double bar inthe other case. In keeping with our previous terminology, we will refer to crosses and doublebars collectively as bridges. In this way, a collection of crosses and double bars connect polesat various heights. Associated to this system is an altered cyclic-time random meander, denotedby X u , T , which is governed by similar rules as its precursor, with the behaviour of the meanderwhen a cross is encountered being the same as when a bridge was encountered in the existingmodel. The difference is that, on jumping over a double bar, unit speed motion along the newpole occurs in the opposite direction to that adopted by the meander immediately before the jump;see Figure 2. As before, the model is well defined for graphs with bounded degree. When u = G is rooted, we will call the meander X u , T recurrent if, when X is begun with vertex component equal to the root, the process has probability one to visit itsinitial location at some positive time; in the other case, the meander will be called transient . Fora regular rooted tree (indeed, for any connected graph of bounded degree), these two conditions CRITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES v w v w F igure
2. An illustration of a cross on the left and a double bar on the right, in eachcase connecting two vertices v and w . The arrows indicate the path followed tilltime T by a particle starting at v and initially moving vertically. Note that thoughwe have drawn arrows on the cross/double bar, in the model the particle crosseseither type of bridge instantaneously.are easily seen to be characterised by the almost sure presence, or respectively absence, of aninfinite cycle in the associated random permutation σ T .Bj ¨ornberg and Ueltschi proved that, on a regular rooted tree, each of whose vertices has d offspring, there is a value T c = T c ( u , d ) ∈ ( ∞ ) which verifies T c ( u , d ) = d + − u ( − u ) − ( − u ) d + o ( d − ) (1)such that cyclic-time random meander X u , T is transient when T ∈ ( T c , T c + Ad − ) and recurrentwhen T < T c . Here, the parameter A > d exceeds acertain value that may depend on A . What Bj ¨ornberg and Ueltschi demonstrate, then, is thepresence of a critical value for the transition from recurrent to transient behavior, at least locallynear the value. It remains possible in principle that recurrent behavior may be reestablished as T increases over the putative critical value by an amount whose order exceeds d − .As Bj ¨ornberg and Ueltschi have noted, the coefficient of d − in (1), viewed as a functionof u , has interesting qualitative similarities with behavior witnessed in numerical studies ofCyTRM ( u , T ) on Z d . The shared features are illustrated in Figure 3: convexity, a minimumin (
0, 1 ) , and a higher value at u = Main result.
As Bj ¨ornberg and Ueltschi did, we will consider the graph G to be a tree whereeach vertex has d offspring. (One result will be valid when each vertex has at least d offspring.)Our main result demonstrates that T c ( u , d ) is indeed the critical point for the transition fromrecurrence to transience. Theorem 1.1. (1)
There exists d ∈ N such that, if G is a rooted tree of bounded degree each ofwhose offspring has at least d offspring, and u ∈ [
0, 1 ] , then there exists T ∈ ( ∞ ) such thatCyTRM(u , T) is transient when T > T . In particular, we may take d = and T = . RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 5 u F igure
3. A plot of [
0, 1 ] → [ ∞ ) : u − u ( − u ) − ( − u ) . The plot bearsqualitative similarities to one obtained numerically in [BBBU15] for the presum-ably critical T -value for CyTRM ( u , T ) on Z d , namely convexity, a unique mini-mum in (
0, 1 ) , and a higher value at 1 than 0.(2) If G is instead chosen to be a rooted tree each of whose offspring has exactly d offspring, withd ≥ and u ∈ [
0, 1 ] , then there exists a T c = T c ( u , d ) such that CyTRM(u , T) is transientfor T > T c and recurrent for T < T c . The critical value T c = T c ( u , d ) satisfies (1); it exceedsd − + d − for d ≥ . Method of proof: a patchwork of four pieces.
Theorem 1.1 is a consequence of four tech-niques of proof that have been employed to investigate the problem. In order to offer an overallorientation to the reader, we summarise these four methods now, all of which have been em-ployed thus far only in the random stirring case when u =
1. We list them roughly in increasingorder for the ranges of T which the methods address. The graph G in question is the regular treewith offspring degree d . I: Absence of large cycles via percolation.
The first argument is very simple. If the pole heightsatisfies T ∈ (
0, log dd − ) , then the probability that a given edge supports either a cross or a doublebar is less than d − , the bond percolation critical value for G . The meander remains among edgesof a single such percolation component and is therefore recurrent. II: Angel’s argument, slightly above the critical value.
Angel specifies a local configurationwhich forces the meander away from the root. More precisely, he defines a local configurationsuch that if a particle encounters it, it will either never return to its current position (so its path istransient), or will move to an offspring vertex where the local configuration has a chance of beingrepeated. Angel proves that the vertices enjoying the local configuration form a super-criticalGalton-Watson tree, which, along with the above claim, gives transience. The local configurationdepends upon there being a low number of bridges on the pole in question; this event only hasa reasonable probability for T slightly above the critical point. As such, the argument can givetransience only for T ∈ [ d − + ( + ε ) d − , 1/2 ] for d high enough (depending on ε ). CRITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES
III: Monotonicity around the critical value.
In [Ham15], it is argued that, for T ∈ ( d − , d − + d − ] , meander transience at T implies transience at any higher value on this interval. In brief,this is accomplished by proving a formula similar to Russo’s formula from percolation theory[Gri99, Theorem 2.25] regarding the effect on the particle’s trajectory of the placement, uniformlyat random, of a single extra bridge on the poles up to distance n from the root. IV: Large cycles, high above the critical value.
In [Ham13], an argument was presented fortransience which works well at high values of T : for example, when d ≥
39 and T ≥ d − . Thisrelies on finding a favourable collection of bridges, called “useful bridges”, whose probability ofoccurrence does not decay with T . The useful bridges serve two purposes: they are locationsfrom where the particle enters unencountered territory (and hence independence comes to theaid of the analysis), and they are also obstacles which the particle must recross back to the rootif it is to not be transient. The bulk of the proof is in establishing a linear rate at which usefulbridges are generated, thereby showing that infinitely many are generated over the course of thetrajectory with positive probability.Bj ¨ornberg and Ueltschi use a different argument in [BU16]. We make no use of this argument,except in order to assert the asymptotic formula (1); the statement of their result is includedbelow for completeness as Proposition 1.3.One of the roles of this article is the adaptation and simplification of argument IV as givenin [Ham13] to the u = d for which theargument as a whole is applicable. We elaborate on both these points in greater detail at the endof Section 2.1.The proof of Theorem 1.1 uses the four arguments mentioned here in a patchwork manner,so that every value T ∈ ( ∞ ) is treated by at least one argument. Our task is to adapt thetechniques to work in the case when u ∈ [
0, 1 ] is not one. Indeed, the next four-part resultindicates the inference that we will respectively make from each adapted argument. Proposition 1.2.
Let G be a rooted infinite regular tree each of whose offspring has exactly d offspring,and X = X u , T be CyTRM on G with parameters u ∈ [
0, 1 ] and T ∈ ( ∞ ) . Then: (1) If T ∈ (
0, log dd − ) , X is recurrent. (2) If d ≥ and T ∈ [ d − + d − , 4 d − ] , or if d ≥ and T ∈ [ d − , 1/2 ] , then X is transient. (3) Let d ≥ and let T , T ′ be such that d − < T < T ′ ≤ d − + d − . If X u , T is transient then sois X u , T ′ . (4) If d ≥ and T ∈ [ ∞ ) , then X is transient. This remains true if we relax our hypothesis toevery vertex of G having at least d offspring. Proof of Theorem 1.1.
Part (1) follows immediately from Proposition 1.2 (4). For part (2), wesimply write out the ranges guaranteed by Proposition 1.2 and check that they overlap.
RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 7 • From Proposition 1.2 (1), we get recurrence for T < log dd − , which implies the same for T ≤ d − + d − . • From Proposition 1.2 (2), we get transience for d ≥
56 and T ∈ [ d − + d − , 4 d − ] ,as well as for d ≥ T ∈ (cid:2) d − , (cid:3) . • From Proposition 1.2 (4), we get transience for d ≥
16 and T ∈ [ ∞ ) . • The range excluded by the above three bullet points is ( d − + d − , d − + d − ) . Nowif d ≥
26, Proposition 1.2 (3) gives monotonicity in ( d − , d − + d − ] . Thus if we furtherhave d ≥
56 for the second bullet point to apply, the existence of a critical T c > d − + d − is implied.The claim regarding the asymptotic formula of T c is a just the formula from the result ofUeltschi and Bj ¨ornberg in [BU16] referenced above. We include it as the next proposition withoutproof for the reader’s convenience. This completes the proof of Theorem 1.1. (cid:3) Proposition 1.3 (Theorem 1.1 of [BU16]) . Let A > be given. Then there exists a d , possiblydepending on A, such that for d ≥ d , there exists a T c = T c ( u , d ) with the property that CyTRM(u , T)is transient for T ∈ ( T c , T c + Ad − ) and recurrent for T < T c . Furthermore, T c ( u , d ) satisfies (1) . It remains, of course, to prove Proposition 1.2. Leaving aside the trivial first argument, thework of adapting arguments II and III is straightforward. We will not rewrite these argu-ments, but rather indicate the necessary changes to the original papers in the final Section 3.Proposition 1.2 (4) entails more substantial adaptation of the argument given in [Ham13]. Wechoose to present a self-contained proof of this result, and do so next, in Section 2.We conclude this section by stating and proving a proposition which will be required in proofsof several parts of Proposition 1.2. It states that the particle cannot move both vertically up anddown (at different times) on any portion of a pole, in spite of the direction-switching double bars.
Proposition 1.4.
Suppose a particle performing
CyTRM ( u , T ) on a tree is at the position ( v , t ) ∈ V × [ T ) and moving upwards. Then at any future time at which it is again at ( v , t ) it will be movingupwards, i.e. it cannot be present at the same location moving in the opposite direction. Proof.
Suppose to the contrary that the particle starting at ( v , t ) and moving up returns to ( v , t ) while moving down after tracing out some path. This means that the path between the two visitsto ( v , t ) is finite.Let ( e , t ′ ) be the first bridge encountered from cyclic motion upwards from ( v , t ) , connectingto ( v ′ , t ′ ) . Observe that for the particle to be at ( v , t ) and moving downwards later, it must CRITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES necessarily come by travelling across ( e , t ′ ) and then travelling downwards. Thus in our pathwe have a pairing between two trips across the bridge ( e , t ′ ) . Now observe that we have anotherpath from ( v ′ , t ′ ± ) (depending on whether ( e , t ′ ) was a cross or a double bar) to itself with theinitial and final directions opposite. By induction, we thus have that every bridge in the originalpath was traversed an even number of times.If particular, double bars were traversed an even number of times, implying that the directionat ( v , t ) finally must be the same as it was initially, a contradiction. (cid:3) Acknowledgment.
The first author thanks Daniel Ueltschi for useful discussions.2. T ransience for high T for u ∈ [
0, 1 ] We now turn to the main technical element of this paper, the proof of Proposition 1.2 (4). Ourargument bears strong similarities to that given in [Ham13], e.g., it uses the same notion of auseful bridge. However, the proof given here, apart from applying even when u =
1, is alsosimpler in certain ways. The differences between our argument and [Ham13]’s will be discussedat the end of the proof outline below.2.1.
Outline of Proof.
We are trying to prove that for sufficiently large T , CyTRM ( u , T ) escapesto infinity with positive probability. This is of course true if T = ∞ , as the process is then justsimple continuous-time random walk on a tree. Here the problem is that we do not necessarilyhave that each move the particle makes is independent of the past; if the particle returns to aportion of the environment it has already visited, its motion will be “deterministic” in that it isdetermined by the past.However, each time the particle moves into unvisited territory it gets a new lease of indepen-dence which we can exploit. Our approach is to show that, with high probability, these “frontierdepartures” (Definition 2.3) into new territory occur often enough, and detrimental returns toexplored territory can be controlled. For the analysis of the occurrence of frontier departureswe need the notion of useful bridges (Definition 2.2) till a given time t , defined according to theparticle’s trajectory up to time t . An important property of these bridges is that their supportingedges have been crossed only once up till time t , and so by the tree geometry, if the particle is tobe recurrent it must recross all edges supporting useful bridges on its journey back to the root—but useful bridges will be defined such that when the particle has the opportunity to make sucha recross, it may instead make a frontier departure into new territory, an event whose probabilityis bounded below in Lemma 2.5.If such a favourable frontier departure occurs, Lemma 2.6 identifies an event which leads to afixed number of useful bridges being encountered immediately after, and gives a lower boundon its probability. If the particle’s trajectory is not so favourable and a frontier departure is notmade, Lemma 2.7 limits the damage done by bounding how many useful bridges can be lost,roughly speaking. Our proof will conclude by showing that overall the number of useful bridges RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 9 grows to infinity with positive probability, which implies that the particle escapes to infinity withpositive probability. This is done by a comparison of the number of useful bridges with a suitablerandom walk on Z which is made to escape to + ∞ .In adapting certain arguments from [Ham13], we make use of Proposition 1.4 in the proof ofLemma 2.5.Apart from adaptations, our proof also differs from [Ham13] in ways that result in a shorterand simpler argument. Notably, our lower bound on the number of useful bridges is obtainedby introducing the move-forward event MF N , T that the particle moves away from the root N timesconsecutively within a time span of T (in this interval we are guaranteed independence no matterthe motion). This is a simple event which streamlines the analysis. In the proof of Proposition 1.2(4), we will be finding a lower bound on the probability of the occurrence of MF N , T in theimmediate aftermath of a frontier departure. If the event occurs, the delay after the frontierdeparture at which it is confirmed to do so is a stopping time. The counterpart to this stoppingtime in [Ham13] was a deterministic duration; this convenient new use of randomness is a sourceof simplification.In terms of results, we obtain transience for d ≥
16 and T ≥ d ≥
39 and T > d − ; however, as Remark 2.10 observes, we may also get a similar range forhigher d by picking parameters T and N differently.2.2. Proofs.
Throughout this section our graph G is an infinite tree where each vertex has at least d offspring. We start by establishing some notation. Notation.
Given the parameter T > P T will denote the probability measure with respect towhich the rate one Poisson process on V × [ T ) is defined.We will refer to a particle being on the pole of a vertex v ∈ V at a height t ∈ [ T ) by thecoordinates ( v , t ) . We similarly refer to a bridge supported by an edge e ∈ E at height t ∈ [ T ) as ( e , t ) .The CyTRM( u , T ) process started at ( φ , 0 ) , where φ is the root, will be denoted by X , so that X ( t ) ∈ V × [ T ) is the position of the particle at time t ∈ [ ∞ ) and X ( ) = ( φ , 0 ) . Y will denotethe projection of X onto the vertex set V , so that Y ( t ) is the vertex whose pole X ( t ) is at. We willadopt the intuitive notation that for any t > X [ t ) = { X ( s ) | s ∈ [ t ) } ,with the obvious analogue for Y .We use notation for two types of hitting times for A ⊆ V × [ T ) : H A = inf { s ≥ | X ( s ) ∈ A } and H t , A = inf { s ≥ t | X ( s ) ∈ A } .Further, if A = { x } , we will replace A in the above notation by x . For an edge e connecting vertices, e + will denote the vertex e is incident to which is closer tothe root and will be called the parent vertex of e . Similarly e − will denote the incident vertexfurther from the root, called the offspring vertex of e . We will refer to the parent vertex of anygiven vertex of the tree to mean the neighbour closer to the root, and likewise for offspringvertices. The graph distance metric will be denoted by dist.We record a simple observation regarding the conditional distribution of the unexplored envi-ronment given the trajectory up to time t . We will make use of this lemma without comment inthe sequel. Lemma 2.1.
Let t > . Consider the law P T given X : [ t ] → V × [ T ) . Let Found t ⊆ E × [ T ) denote the set of bridges that X has crossed during [ t ] , and let UnExplored t ⊆ E × [ T ) be all elementsof E × [ T ) neither of whose endpoints lie in X [ t ] . Then the distribution of the collection of bridges B given X [ t ] is given by Found t ∪ B ( t , ∞ ) , where B ( t , ∞ ) is a collection of bridges distributed as a Poissonprocess on E × [ T ) with intensity UnExplored t with respect to product Lebesgue measure. Proof.
It is obvious that Found t is contained in B as it is known given X [ t ] . From the indepen-dence property of Poisson processes it follows that the distribution of the remaining bars, i.e.those in UnExplored t , is unaffected. (cid:3) Definition 2.2 (Useful bridges) . We define, for t >
0, a set U t ⊆ E × [ T ) of useful bridges attime t . A bridge ( e , s ) ∈ E × [ T ) belongs to U t if • H e + < H e − < t , • H e − − H e + < T /2, • { ˜ t ∈ [ t ] : Y ( ˜ t ) = e + } = [ H e + , H e − ) , and • { ˜ t ∈ [ t ] : Y ( ˜ t ) = e − } is an interval with right endpoint strictly less than t .Thus, a bridge is useful at time t if it has been crossed before that time, the particle has spentat most time T /2 at the bridge’s parent vertex, has visited the parent and offspring vertices onlyonce, and is not at the offspring vertex at time t . Definition 2.3 (Frontier time) . A time t > frontier time if Y ( t )
6∈ { Y ( s ) | ≤ s < t } ,i.e., the vertex whose pole X ( t ) is at has not been visited before time t . Definition 2.4 (Frontier departure) . Under P T given X : [ t ] → V × [ T ) , if ( e , s ) ∈ U t andconditional upon H t , e − < ∞ , we say X makes a frontier departure from e if after time H t , e − , at themoment of departing { e + , e − } , X arrives at the pole of a vertex it has not visited before.Note carefully that in the above definition we are considering the moment of departure from { e + , e − } , and not from e − alone; so the particle may go from e − to e + first and then depart to anunvisited vertex as part of a frontier departure. RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 11
Lemma 2.5.
Let t > . Consider the conditional distribution of P T given X [ t ] . Let ( e , s ) ∈ U t withe + = φ be chosen measurably with respect to X [ t ] , and condition further on H t , e − < ∞ . Then theprobability of making a frontier departure is at least d − d + ( − e − ( d − ) T /2 ) . Proof.
Since ( e , s ) ∈ U t , we have that e − has been visited by time t . By the conditioning that H t , e − < ∞ , we note that on the particle returning to the pole at e − , it can either stay on the poletill reaching ( e , s ) (in which case it will jump back to e + ), or it can jump to another vertex beforereaching the bridge ( e , s ) . Let J be the jump event that jumping to e + occurs, and let J c be thecomplementary event.To simplify notation, let τ be the hitting time of { e + , e − } c after time t , i.e., τ = H t , { e + , e − } c .We analyse the case where J c occurs first. Since ( e , s ) ∈ U t , H t , e − is the time of first return of theparticle to the pole at e − . So by Proposition 1.4, the particle is travelling in an unexplored portionof the pole. Further, since J c occurring is equivalent to there being a bridge on e − different from ( e , s ) , we only need consider where it connects to: obviously there are d − d + P T (cid:16) Y ( τ ) Y [ τ ) | X [ t ] , J c (cid:17) ≥ d − d + J occurs, i.e., the particle travels back to e + via the bridge ( e , s ) . Again by thedefinition of U t and Proposition 1.4, the particle travels in the direction of unexplored area on thepole at e + . The unexplored portion of the pole has length at least T /2 since ( e , s ) ∈ U t impliesthe explored interval has length at most T /2. So, conditioned on there being at least one bridgein this unexplored portion, we need to consider the probability that it connects to an unexploredvertex. Doing so and multiplying by the probability of the conditioning event, P T (cid:16) Y ( τ ) Y [ τ ) | X [ t ] , J (cid:17) ≥ d − d + (cid:16) − e − ( d + ) T /2 (cid:17) .Combining the above two gives the lemma. (cid:3) In the next lemma we define the move-forward event MF N , T described earlier and get a lowerbound on its probability. This event is the main source of simplification of our proof in com-parison to [Ham13]. Though in some sense it is quite a crude event, its job is to generate usefulbridges, and it turns out that this is sufficient for our purpose. Lemma 2.6.
Let X be a
CyTRM ( u , T ) started at ( v , t ) and MF N , T be the move-forward event that theparticle goes forward at least N times consecutively in the time interval ( T ) . On MF N , T , let τ be therandom time at which the N th consecutive bridge is crossed, i.e., τ = inf { s ≥ | dist ( Y ( s ) , v ) = N } .Then we have P T ( MF N , T ) ≥ (cid:18) − d + (cid:19) N " − e N − ( d + ) T (cid:18) ( d + ) TN (cid:19) N = : p ( ) N , T , d . Further, on the event MF N , T , |U τ | ≥ N − a.s. Also, if we condition on t being a frontier time and onX [ t ] , the above event with the time interval ( T ) replaced by ( t , t + T ) occurs with the same probability,and on that event |U τ t | ≥ N − a.s., where τ t = inf { s ≥ t | dist ( Y ( s ) , Y ( t )) = N } . Proof.
Recall that the gap distribution of a Poisson process of parameter d + d + ξ , . . . , ξ N be iid Exp( d +
1) random variables. By iteratively conditioning on moving forward onestep and using the independence obtained by moving forward (regardless of vertical direction ofmotion), we obtain P T ( MF N , T ) ≥ (cid:18) − d + (cid:19) N · P ( ξ + . . . + ξ N ≤ T ) . (2)We need an upper bound on P T ( ξ + . . . + ξ N > T ) . Exponentiating, using the Markov inequality,and recalling that the moment generating function of Exp( d +
1) is given by f ( λ ) = d + d + − λ , weget P T ( ξ + . . . + ξ N > T ) ≤ e − λ T (cid:18) d + d + − λ (cid:19) N .This is minimised when λ = d + − N / T , which gives P T ( ξ + . . . + ξ N > T ) ≤ e N − ( d + ) T (cid:18) ( d + ) TN (cid:19) N .Substituting back in (1) implies the claimed lower bound. From the definition of U τ , it followsthat the last bridge is not in U τ as the particle has not yet left the offspring vertex of the lastbridge. Of the remaining N − T /2 time at the offspring vertex before jumping; however, this can happen at most once ina time interval of length T .The fact that the same is true in the time interval ( t , t + T ) when conditioned on t being afrontier time is straightforward, since the particle is in unexplored territory. More precisely,conditional on X [ t ] and t being a frontier time, Lemma 2.1 implies that the distribution of thebridge locations on the pole of Y ( t ) remains unchanged, and so the above argument appliesdirectly. (cid:3) The next lemma considers the situation where the particle returns and does not make a frontierdeparture, so that Lemmas 2.5 and 2.6 do not apply. Its role is to control the damage in thissituation by bounding the number of useful bridges, which may be viewed as obstacles to theparticle returning to the root, that can be undone.
Lemma 2.7.
Let t > , and e ∈ U t be the bridge last crossed in X [ t ] . Let p ( e + ) be the parent of e + .Then, conditionally on e + = φ and H t , p ( e + ) < ∞ , we have that |U t \ U H t , p ( e +) | ≤ a.s. Proof.
We write U t for the set of edges that support a bridge in U t . Note that for each t > U t : no two elements in U t can be supported on the same edge. Hence, it suffices to derive the statement of the lemma with U t replaced by U t . RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 13
The fourth requirement in the definition of U H t , p ( e +) and U t gives that the only way an edge f ∈ U t will not be in U H t , p ( e +) is if the particle visits f − in ( t , H t , p ( e + ) ) . Note that H t , p ( e + ) is notincluded. Now by the tree’s geometry, the only edges of U t whose children may be visited in ( t , H t , p ( e + ) ) are e and ( p ( e + ) , e + ) . Thus the lemma follows. (cid:3) Definition 2.8 (Acceptable return) . Let t > ( e , s ) belong to U t . Write e c : = V ( G ) \ { e + , e − } . If X returns to e after time t , i.e., H t , e − < ∞ , we say the return is acceptable if(1) X makes a frontier departure from e (i.e., X leaves to a previously unencountered vertex),say at time τ (so that τ is a frontier time);(2) and X goes N steps forward consecutively in the time interval ( τ , τ + T ) (note that byright-continuity of X the frontier departure step cannot be counted towards N ). Remark 2.9.
Observe that, conditional on X making a frontier departure, Lemma 2.6 says thatitem 2 above occurs with probability at least p ( ) N , T , d . Thus we can combine Lemmas 2.5 and 2.6 tofind that, conditional on H t , e − < ∞ for some bridge ( e , s ) ∈ U t , P T ( return to e is acceptable | H t , e − < ∞ ) ≥ d − d + ( − e − ( d − ) T /2 ) × p ( ) N , T , d = : p ( ) N , T , d .Now we may prove Proposition 1.2 (4). The idea of the proof is to estimate the number ofuseful bridges at a sequence of random stopping times which we will construct. If we can showthat with positive probability this number goes to infinity without ever hitting 0, then the CyTRMwill be seen to be transient—the particle cannot have returned to the root.We do this by using Lemma 2.6, i.e., by considering the event of moving forward consecutively N steps after a frontier departure; each time this occurs, the number of useful bridges at thetime of completing the N th step increases by at least N −
2. Similarly we control the effectof bad returns by Lemma 2.7. Using the bounds on the probabilities of these events, we canstochastically dominate |U t | by a random walk on Z with related transition probabilities. Thefinal step is to analyse for what values of T , d , and N this random walk can be guaranteed tohave positive drift, for such a random walk will stay positive forever with positive probability.We now turn to the technical details. Proof of Proposition 1.2 (4).
We construct a sequence of stopping times τ k where we estimatethe number of useful bridges. The stopping times will be defined iteratively based on when theparticle next returns to the child e − of a useful bridge e (if it does) and where it jumps to fromthere. If the return is acceptable, i.e., on returning it moves forward into new territory and thengoes N steps forward consecutively, we will have the next stopping time be the moment that itcompletes the N th step. If the return is not acceptable, we will have the next stopping time bewhen the particle reaches p ( e + ) (the worst case), if it does.Throughout we will need the number of useful bridges to be at least two; this is only a technicalrequirement to ensure that we have at least one useful bridge not joined to φ , as Lemmas 2.5 and 2.7 assume the parent vertex is not the root. Hence if at any point |U t | <
2, we choose togive up.Define τ = inf { t ≥ |U t | ≥ } . Observe that τ < ∞ with positive probability; otherwise,set τ j = − ∞ for every j >
1, as a technical convention to say that we have failed and are givingup. Similarly for k ∈ N + , if |U τ k | ≤
1, set τ j = − ∞ for all j > k .Otherwise, denoting the last bridge crossed in U τ k before time τ k by e k , define χ k = H τ k , e − k . Thistime is when the particle next returns to e − k . Now there are a few cases (dist is the graph distanceon G ): • If χ k = ∞ , set τ j = ∞ for j > k , as a technical convention to indicate success; the particle’strajectory is transient. • If χ k < ∞ and the return of X to e k is acceptable, set τ k + = inf { t ≥ H χ k , e ck | dist ( Y ( t ) , e − k ) = N } ,i.e., the first time after H χ k , e ck (the frontier departure time) that X reaches N steps forward. • If χ k < ∞ and the return of X to e k is not acceptable, set τ k + = H χ k , p ( e + ) ,which may be infinite, in which case the particle’s trajectory is transient.For k ∈ N , define u k = |U τ k | . Specify three random variables p k , q k , r k , defined under P T given X [ τ k ] , as follows. Let A k be the event that the return to e k is acceptable, and set p k = P T (cid:16) χ k < ∞ | X [ τ k ] (cid:17) q k = P T (cid:16) A k | X [ τ k ] , χ k < ∞ (cid:17) r k = P T (cid:16) H χ k , p ( e + ) < ∞ | X [ τ k ] , χ k < ∞ , A ck (cid:17) .Note that u k is measurable with respect to X [ τ k ] . Note also that, by the definition of anacceptable return and Lemmas 2.6 and 2.7, the conditional distribution of u k + − u k given X [ τ k ] and { u k > } stochastically dominates the law ( − p k ) · δ ∞ + p k q k · δ N − + p k ( − q k )( − r k ) · δ ∞ + p k ( − q k ) r k · δ − ,which is parametrized by ( p k , q k , r k ) . Since it is not in our favour if p k = r k =
1, this lawstochastically dominates the one where p k , r k = P T ( · | X [ τ k ] , u k > ) -a.s., q k ≥ p ( ) N , T , d .In summary, conditional on X [ τ k ] and { u k > } , the law of u k + − u k P T -a.s. stochasticallydominates the law p ( ) N , T , d · δ N − + ( − p ( ) N , T , d ) · δ − . RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 15
Now let Q : N + → R denote the random walk with independent increments whose law is p ( ) N , T , d · δ N − + ( − p ( ) N , T , d ) · δ − and initial condition Q ( ) =
2. Let ρ be the first time Q goesstrictly below 2, and define Q ∗ : N + → R by Q ∗ ( i ) = Q ( i ) if i ≤ ρ i > ρ for each i ∈ N + .We now have that conditionally on τ < ∞ , { u i : i ∈ N + } stochastically dominates { Q ∗ ( i ) : i ∈ N + } . Thus we need to find N , T , d such that with positive probability Q ( i ) tends to infinitywhile staying strictly above 1 always. This is satisfied if the drift is positive, which is to say thatit suffices to have0 < ( N − ) p ( ) N , T , d − ( − p ( ) N , T , d )= N p ( ) N , T , d − = N d − d + ( − e − ( d − ) T /2 ) (cid:18) − d (cid:19) N " − e N − ( d + ) T (cid:18) ( d + ) TN (cid:19) N −
2. (3)Note that (3) is increasing in T for T > N ( d + ) − ; hence if some choice of ( N , T , d ) works in thisrange, so will higher values of T . Now taking N = T = d ≥
16 gives a positive driftby direct calculation. Therefore u i remains above 1 and goes to infinity with positive probabilityfor N = T ≥ d ≥
16, which can occur only if the particle’s trajectory is transient,thus proving the theorem. (cid:3)
Remark 2.10.
Note that it was not necessary to take N = T ≥ ( T c , 0.5 ] is covered by the other parts ofProposition 1.2. In fact, we can fix ε >
0, take T = ( + ε ) N ( d + ) − , and then adjust N and d inorder to make (3) positive. It is easy to check that this is possible by taking N and d sufficientlylarge. In other words, given an ε >
0, there exist high N and d such that CyTRM ( u , T ) is transientfor T > ( + ε ) N ( d + ) − .We end this section by indicating which of these lemmas and propositions have been taken ormodified from [Ham13]. Our Lemma 2.1 is Lemma 2.1 of [Ham13], our Lemma 2.5 is Lemma 2.6of [Ham13], and our Lemma 2.7 is Lemma 2.14 of [Ham13].3. M odifications of P revious P roofs In this section we show how to modify existing proofs, namely from [Ang03] and [Ham15], tocomplete the proofs of part (2) and (3) of Proposition 1.2. Angel’s proof in [Ang03] is applicableto u = u =
1, as elaborated in the appendix of [BU16], but the bounds can be significantly tightened by a more careful calculation. This isneeded for the bounds we claim.We do not provide self-contained proofs because the involved changes are too mundane towarrant doing so. However, we have provided an overview with indications on how to modifythe original proofs to apply when u =
1. The reader may wish to refer to the original papers toget a complete understanding of the argument.3.1.
Modification of Angel’s Proof.Theorem 3.1.
Let G be a regular tree with d offspring at every vertex (so ( d + ) -regular). Then for any ε > , there exists d such that for d ≥ d and any u ∈ [
0, 1 ] , CyTRM ( u , T ) on G will be transient forT ∈ [ d − + ( + ε ) d − , 1/2 ] . In particular, we have that
CyTRM ( u , T ) is transient for T ∈ [ d − + d − , 4 d − ] for d ≥ and forT ∈ [ d − , 1/2 ] for d ≥ . This is Theorem 3 of [Ang03], but for u =
1. The proof is essentially [Ang03]’s except for aminor modification. In particular, the proof of a claim in the initial section of the proof of [Ang03,Theorem 3] must be modified for u =
1, which we isolate below as Lemma 3.4.First we recall some required definitions from [Ang03]:
Definition 3.2 (Good vertex) . Let v be a vertex and u be its parent. We say v is good if there isonly a single bridge between u and v . Definition 3.3 (Uncovered vertex) . Let v be a good vertex, u be its parent and a good vertex, and v ′ be a sibling of v . Call v covered by v ′ if the bridges from u to v ′ cyclically separate (on the poleof u ) the unique bridge from u to v and the unique bridge from u to its parent.We say a vertex v as above is uncovered if it is not covered by any of its siblings. Lemma 3.4.
Suppose v is uncovered and the particle reaches u, the parent of v. Then either the particlereaches v, or it leaves u at some point and never returns.
Proof.
Suppose that the first case does not occur, i.e., the particle does not reach v . We mustshow that ( u , t u ) is not part of a finite cycle. Assume without loss of generality that the particleis moving up from ( u , t u ) .Since v is uncovered, we have that if a bridge supported on an edge is present in I : = [ t u , t v ] ,there is no bridge supported by the same edge outside I . The fact that we never reach v impliesthat the particle can in future be present at vertex u only in the interval [ t u , t v ] . Due to the treegeometry, this tells us that if ( u , t u ) is part of a finite cycle, the bridge at t u must be encounteredvia downward motion at some point. But since we initially moved up from ( u , t u ) , Proposition 1.4tells us that this is not possible. (cid:3) Proof sketch of Theorem 3.1.
Given Lemma 3.4, two things are needed to complete the proof. Itmust be shown that the number of good, uncovered offspring of distinct vertices are independent
RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 17 random variables, so that the good, uncovered connected component containing φ is a Galton-Watson tree, and then it must be shown that for the claimed choices of d and T , this randomvariable has mean greater than 1—as this is equivalent to the Galton-Watson tree containing withpositive probability an infinite path starting at the root.This is accomplished by the rest of Angel’s proof, which goes through even for u =
1, andso we obtain the same result as there. Also refer to [Ham13, Lemma B.2] for the proof of theparticular bounds on d and T -intervals. (cid:3) Proof of Proposition 1.2 (2).
This is the same statement as the second part of Theorem 3.1. (cid:3)
Modification of Hammond’s Proof of Monotonicity.
As substantiated in the appendixof [BU16], Hammond’s proof of monotonicity given in [Ham15] works essentially without mod-ification for any u ∈ [
0, 1 ] . However, the bound on d obtained in that paper for which the resultapplies can be easily tightened to get the bounds we claim by a closer examination of the proofand some new calculations. In this subsection we discuss how to go about doing this.We do not provide a self-contained proof of the monotonicity claimed in Proposition 1.2 (3).Instead, we give a sketch, taking certain claims and inputs from [Ham15] as black boxes. Areader who has not read [Ham15] should be able to read this subsection and obtain a clear ideaof the overall proof, modulo certain facts which are stated but not proved here. To obtain acomplete proof, it is recommended to the reader to refer to [Ham15] alongside this subsection toget a detailed understanding of where and how the original proof and calculations are modified.First we recall some notation from [Ham15]. The idea of that paper is to consider the probabil-ity that CyTRM ( T ) does not return to the pole over the root φ , denoted p ∞ ( T ) , and to show that itis non-decreasing in the interval ( d − , d − + d − ] which contains the critical point. This is doneby considering “local approximations” p n ( T ) , defined as the probability that CyTRM ( T ) everreaches level n of the tree, and proving that these functions are differentiable and non-decreasingin the required interval. This is clearly sufficient as p n ↓ p ∞ pointwise. Unlike in [Ham15], herethe quantities p n and p ∞ are defined with respect to CyTRM ( u , T ) instead of CyTRM ( T ) . Proof of Proposition 1.2 (3).
By the above discussion, this is implied by the next proposition. (cid:3)
Proposition 3.5 (Modification of Proposition 1.8 of [Ham15]) . Let d ≥ and suppose d − < T ≤ d − + d − . Then for each n ≥ , p n is differentiable at T and d p n d T ( T ) > . Remark 3.6.
Proposition 1.8 of [Ham15] is used to prove that paper’s Proposition 1.3 (ourProposition 1.2 (3)), and is stated as d p n d T ( T ) > d e − Td p n ; however, as indicated above, the proofof Proposition 1.3 itself only requires d p n d T ( T ) >
0, which is partly what allows us to get a betterrange for the d where Proposition 1.2 (3) is applicable. Notation.
We let T n be the subgraph induced by the vertices within distance n of the root. Using a formula analogous to Russo’s formula from percolation theory, we can write the de-rivative of p n in terms of certain “pivotal” events P + n and P − n . These events are defined in termsof the effect of a “uniformly added bridge”. To be precise, to the existing random arrangementof bridges, we add one additional bridge A n sampled from normalized Lebesgue measure on E ( T n ) × [ T ) , independently of the existing bridges, which is direction maintaining with prob-ability u and direction switching with probability 1 − u .This new bridge A n can potentially affect the trajectory of the particle. One of three things canhappen: the trajectory of the particle originally did not exit T n , and now does; the particle didexit T n originally, but no longer does; or finally, the event of the particle exiting T n is unaffected.We denote by P + n and P − n the events that the first and the second possibilities occur.We can now state (without proof) an expression for the derivative of p n in terms of theseevents. Lemma 3.7 (Lemma 1.7 of [Ham15]) . For each n ∈ N , p n : ( ∞ ) → [
0, 1 ] is differentiable; for T > , d p n d T ( T ) = | E ( T n ) | (cid:0) P T ( P + n ) − P T ( P − n ) (cid:1) .The probability P T ( P + n ) − P T ( P − n ) is decomposed as A + A , where A = P T ( P + n ∩ C ∩ B c ) − P T ( P − n ∩ C ∩ B c ) , A = P T ( P + n ∩ C ∩ B ∩ N ) − P T ( P − n ∩ C ∩ B ∩ N ) .Here C is the crossing event that the particle reaches A n before exiting T n . If C occurs, B is the bottleneck event that some edge between the root and the parent vertex supporting A n supports asingle bridge. Let b n be such a bridge that is farthest from the root. Suppose both C and B occur,and that the particle’s trajectory is periodic. Then it must cross back along b n after crossing itthe first time. The no escape event N is the event that the particle, considered from the time itrecrosses b n , reaches the root before exiting T n .These details are provided to give the reader some idea of the original proof and to be consis-tent with the notation in [Ham15]; we will not actually be needing these details for our modifi-cations.At this point our next step is to give a lower bound on A . This is the content of the next twolemmas. The first is stated without proof, but the second is one where we will need to make amore careful calculation than in [Ham15]. Notation.
For convenience, we write τ = Td , so that we are interested in τ ∈ [
1, 1 + d ] . Lemma 3.8 (Lemma 4.3 of [Ham15]) . Suppose that n ≥ , d ≥ , and T > . Then P T ( P + n ∩ C ∩ B c ) ≥ de − τ p n − | E ( T n ) | . RITICAL POINT FOR INFINITE CYCLES IN A RANDOM LOOP MODEL ON TREES 19
Lemma 3.9 (Modification of Lemma 4.5 of [Ham15]) . Suppose that n ≥ , d ≥ , and ≤ τ ≤ + d. Then P T ( P − n ∩ C ∩ B c ) ≤ p n − | E ( T n ) | (cid:18) τ ( τ + ) + e − a ( a − a − )( − a ) (cid:19) , where a = τ e + τ d d ( e − ) . Proof sketch.
Here we indicate how to modify the proof of Lemma 4.5 in [Ham15] to obtain ourclaim. This comes down to explicitly evaluating a certain sum instead of bounding it. In theproof of [Ham15, Lemma 4.5], the following inequality is obtained: P T ( P − n ∩ C ∩ B c ) ≤ ∞ ∑ k = A n , k ,where A n , k is a technical quantity which we will not define; for the purpose of getting a tighterbound, note that it is proven in [Ham15] that A n ,0 ≤ p n − | E ( T n | τ ( τ + ) and A n , k ≤ p n − | E ( T n ) | e − (cid:16) ( − e − τ ) − e τ / d + τ d − (cid:17) k ( k + ) .Now we only need to estimate the sum of the right hand side as k varies from 0 to ∞ . This isdone in [Ham15] by bounding the exponential term using τ ≤ ( k + ) ≤ k + , but amuch tighter bound is easily obtainable.Using that τ ≥
1, we can bound the exponent as τ e + τ / d d ( − e − τ ) ≤ τ e + τ d d ( e − ) = : a .The identity ∑ ∞ k = ( k + ) a k = a ( a − a + ) / ( − a ) is valid for any − < a <
1. With thisand the bound on the exponent, summing the series gives the claimed upper bound. (cid:3)
In [Ham15], a lower bound on A is obtained from the bound on A ; that argument goesthrough even when u =
1. Hence, we obtain the following:
Lemma 3.10 (Proposition 3.2 of [Ham15]) . Let n ≥ , d ≥ , τ ∈ [
1, 1 + d − ] . ThenA = P T ( P + n ∩ C ∩ B ∩ N ) − P T ( P − n ∩ C ∩ B ∩ N ) ≥ Proof of Proposition 3.5.
Lemma 3.7 asserts thatd p n d T ( T ) = | E ( T n ) | (cid:0) P T ( P + n ) − P T ( P − n ) (cid:1) = | E ( T n ) | ( A + A ) .Using Lemmas 3.8, 3.9, and 3.10, we find thatd p n d T ( T ) ≥ p n − (cid:20) de − τ − τ ( τ + ) − e − a ( a − a + )( − a ) (cid:21) .where a = τ e + τ d d ( e − ) . We see that the expression is increasing in d for fixed τ and decreasing in τ forfixed d . Thus it is bounded below by the value at τ = + d , and numerically it can be verified that this expression (with τ = + d ) becomes strictly positive at d =
26. Hence it is positivefor all d ≥
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