Critical drift estimates for the frog model on trees

Place an active particle at the root of a $d$-ary tree and a single dormant particle at each non-root site. In discrete time, active particles move towards the root with probability $p$ and, otherwise, away from the root to a uniformly sampled child vertex. When an active particle moves to a site containing a dormant particle, the dormant particle becomes active. The critical drift $p_d$ is the infimum over all $p$ for which infinitely many particles visit the root almost surely. Guo, Tang, and Wei proved that $\sup_{d\geq 3} p_d \leq 1/3$. We improve this bound to $5/17$ with a shorter argument that generalizes to give bounds on $\sup_{d \geq m} p_d$. We additionally prove that $\limsup p_d \leq 1/6$ by finding the limiting critical drift for a non-backtracking variant.


Introduction
Let T d be the infinite, rooted d-ary tree in which each vertex has d ≥ 2 child vertices.Place an active particle at the root ∅ and a single dormant particle at each non-root site.Fix p ∈ (0, 1) and have each active particle perform a nearest neighbor p-biased random walk.At each discrete time step an active particle moves one step towards the root with probability p and away from the root to a uniformly sampled child vertex with probability 1 − p.When an active particle moves to a site with a dormant particle, the dormant particle becomes active and begins its own independent p-biased random walk.Call this process the frog model with drift on T d and denote it by FM(d, p).
Frog model dynamics capture aspects of the spread of infection, a rumor, or energy.When the underlying graph is infinite, a basic question is whether or not infinitely many particles visit the root.Many papers have studied this with simple random walks on integer lattices and trees [TW99, AMP02, Pop01, HJJ17, HJJ19, MR19].There has been recent interest in the variant in which active particles perform biased random walk [DGH + 18, BFJ + 19, GTW22].
A root visit is counted each time an active particle moves to the root.Let V t be the number of root visits up to time t and V FM(d,p) := lim t→∞ V t be the total number of root visits.We say that FM(d, p) is recurrent if P(V FM(d,p) = ∞) = 1.Recurrence satisfies a 0-1 law (see [BFJ + 19, Proof of Proposition 1.4]).Accordingly, call the process transient if it is not recurrent.Define p d := inf{p : FM(d, p) is recurrent} to be the infimum over all drifts for which FM(d, p) is recurrent.Since a single p-biased walk is recurrent for p ≥ 1/2, we are only interested in p ∈ (0, 1/2).
A natural case FM(2, 1/3) has active particles performing simple random walk on the binary tree.Hoffman, Johnson, and Junge resolved a longstanding open problem by proving that FM(2, 1/3) is recurrent [HJJ17].Conversely, FM(2, p) with p < 1/3 is transient since the dominating process with all particles initially active is transient.Thus, p 2 = 1/3.
Increasing p creates a stronger drift towards the root, and increasing d results in more dormant frogs.Both of these effects should result in more visits to the root.An intriguing feature of FM(d, p) is that there is no known proof that V FM(d,p) stochastically increases in p or d.For example, it is not obvious that p 2 = 1/3 is a uniform bound for p d .The main result from [BFJ + 19] proved a weaker bound sup d≥3 p d ≤ 0.4155.In [GTW22], Guo, Tang, and Wei improved this bound to 1/3.Our first result is a slightly better bound.
Besides the bound improvement, we see several positive consequences of Theorem 1.One is that the proof uses a different technique than what was used in [GTW22], which provides new perspective.A particular highlight is a simple to check criteria for recurrence of FM(d, p) in Proposition 10.Another nice consequence is that our proof is shorter than that given in [GTW22] (four versus nineteen pages).The third is that the bound we obtain is strictly less than p 2 = 1/3.This implies that p d < p 2 for all d ≥ 3, which supports the conjecture from [BFJ + 19] and [GTW22] that p d is decreasing.The fourth positive consequence is that our technique can be generalized to give better bounds on sup d≥m p d as m is increased.See Remark 13 for more details.It is unclear if the approach from [GTW22] could be as easily generalized.To illustrate how the generalization goes, we prove an extension for d = 4.Here q * is the critical drift for the branching p-biased random walk in which each particle does not branch when moving towards the root (which it does with probability p) and splits into two particles when moving away from the root.Our second result is a limiting bound on p d that is near q * .Theorem 3. lim sup d→∞ p d ≤ 1/6 ≈ 0.1667.
Intuition suggests that as d becomes larger, most steps away from the root by particles in FM(d, p) will be to sites containing dormant particles.Thus, FM(d, p) ought to converge to this branching random walk as d → ∞.As mentioned previously, monotonicity of p d has yet to be established.So, both the existence of the limit and convergence to q * remain open.
The only known monotonicity result for FM(d, p) is [BFJ + 19, Proposition 1.2], which states that V FM(d,p) V FM(kd,p) for any positive integer k.One difficulty is that the frog model has regimes in which the set of sites visited by active particles contains a linearly expanding ball centered at the root [HJJ19].No activation occurs in this growing region.This distinguishes the frog model from branching random walk on a macroscopic level, and casts a shadow of doubt on (1).
Theorem 3 is proven by exactly computing the limiting critical drift for the non-backtracking frog model denoted by nbFM(d, p).This is a relevant model since all arguments that we know of for recurrence of FM(d, p) rely on proving that nbFM(d, p) is recurrent.In nbFM(d, p), paths of active particles are nonbacktracking.Let Initially, there is one active frog at the root.It moves to a uniformly sampled child vertex in the first step and activates the dormant frog there.Just activated frogs move towards the root with probability p * , and otherwise away from the root to a uniformly sampled child vertex.For subsequent steps, if the previous step was towards the root, then the next step will be towards the root with probability p.If the previous step was away from the root, all subsequent steps will be away from the root to uniformly sampled child vertices.Any particles that visit the root are killed there and no longer participate in the process.Let V nbFM(d,p) denote the total number of root visits in nbFM(d, p) and say that the process is recurrent if We find the exact limiting value of p d .
Theorem 4. lim d→∞ p d = 1/6.Theorem 4 is used to derive Theorem 3. Theorem 4 is a valuable contribution in and of itself since it suggests the truth of (1).Indeed, the intuitive limit nbFM(∞, p) is a branching process nbBRW(p) in which particles move towards the root with probability p and do not branch for some geometric distributed number of steps, after which they move away from the root branching into two particles at each step.It follows from Lemma 15 that nbBRW(p) has critical drift 1/6, thus p d converges to its intuitive limit.Note that nbFM(p) also exhbits a linearly expanding ball of visited sites when the initial particle density is high enough [HJJ19].So, the "shadow of doubt" mentioned earlier from this macroscopic effect does not seem to effect convergence of the critical drift.
Another benefit of of Theorem 4 is that it provides useful guidance on where not to direct future efforts towards establishing (1).All proofs that we know of for recurrence of a frog model on an infinite tree did so by proving that a nonbacktracking sub-process is recurrent.Since q * < 1/6, our result suggests that any argument using a non-backtracking frog model will fall short of proving that p d → q * .Some new type of argument that engages directly with FM(d, p) appears to be needed.
The arguments we employ to upper bound p d and p d use approximations to the frog model that are less recurrent.To get a sense of how much precision is lost, we conducted some numerical simulations to estimate p 3 , p 3 , and p 4 .We found that p 3 ≈ 0.25; p 3 ≈ 0.2725; p 4 ≈ 0.246.(3) Details are in Section 6.In [HJJ17], it was conjectured that FM(3, 1 4 ) i.e., the frog model with simple random walks, is recurrent.So, under this assumption p 3 = 1 4 .Our data gives more support to this conjecture (see Figure 2).The values of p 3 and p 4 are within about .02 of the bounds from Theorem 1 and Theorem 2. This suggests that the our proofs do not sacrifice much accuracy.It is also interesting to see the (simulated) discrepancy between p 3 and p 3 (about .0225)that results from restricting to non-backtracking random walk paths.
1.1.Overview of proofs.The proof that sup d≥3 p d ≤ 1/3 from [GTW22] followed the blueprint from [HJJ17].The calculations in [HJJ17] were involved, and became much more complex in the generalization in [GTW22].We work with a Poissondistributed number of dormant particles per site.Poisson thinning makes many intricate dependencies vanish.A comparison result from [JJ18] lets us convert our findings back to the one particle per site setting of FM(d, p).
Our main tool is a self-similar frog model SFM(d, p) that embeds in the usual frog model so that it has fewer root visits.We denote by V SFM(d,p) the number of root visits in SFM(d, p).It was observed in [HJJ16] that V SFM(d,p) satisfies a recursive distributional equation in the simple random walk setting.A similar equation holds for arbitrary p.The equation relates V SFM(d,p) to 1 + U thinned independent copies of V SFM(d,p) , where U is the number of leaves visited in a frog model on a star graph (see Figure 1).In Proposition 10, we reduce proving recurrence to finding a stochastic lower bound for U whose Laplace transform satisfies a certain inequality.
Theorem 2 uses the same approach as Theorem 1, and Theorem 3 follows immediately from Theorem 4. The proofs of Theorem 1 and Theorem 4 come down to constructing the right stochastic lower bound for U .For Theorem 1, we modify what occurs on T d to resemble the setting with d = 3.For Theorem 4, we leverage the fact that when the drift is fixed, we do not need many leaves of the star graph to be visited in order to satisfy Proposition 10.The arguments presented are not simple rehashes of past techniques.The stochastic lower bounds are novel and tailored to FM(d, p).See Remark 11 for more about the difficulties.
Another ingredient in the proof of Theorem 4 is connecting nbFM(d, p) with its intuitive limiting multitype branching random walk.This substantial endeavor is a technical contribution.Section 4 defines the multitype branching random walk and then works out its transience and recurrence properties.The main thrust is extending results from [MMP01] to our setting.The transience/recurrence criteria in Lemma 14 are novel and may be of future use for the study of multitype branching random walks.1.2.Organization.In Section 2, we define the self-similar frog model and deduce some of its properties.This culminates with a sufficient condition for recurrence of SFM(d, p) given at Proposition 10.We use this in Section 3 to prove Theorem 1. Section 4 gives transience and recurrence conditions for a multitype branching random walk and relate them back to the frog model.In Section 5, we prove Theorem 3, which has Theorem 4 as an immediate corollary.Finally, in Section 6, we provide some numerical simulations that complement our results.1.3.Acknowledgements.We are grateful to the authors of [HJJ17] whose code formed the basis of the simulations performed in Section 6.We would also like to thank Serguei Popov for sending us an electronic copy of [CMP98] whose result is applied in the proof of Lemma 15.

The self-similar frog model and associated operator
First a few remarks on notation.We abbreviate the Poisson distribution with mean λ by Poi(λ).Given two nonnegative random variables X and X , we say that X is stochastically smaller than X if P(X ≥ a) ≤ P(X ≥ a) for all a ≥ 0. We will denote this by X X .Similarly, given two probability measures π and π on [0, ∞] we say that π π if π((a, ∞)) ≤ π (a, ∞) for all a ≥ 0.
The self-similar frog model operator.Red sites contain particles that are initially active and blue sites contain initially dormant particles.Aπ is the law for the number of particle frozen at ∅ when the process fixates and U is the number of vertices among v 2 , . . ., v d that are ever visited.Empty boxes on the right at v 1 , . . ., v d represent sites whose particles were activated.
2.1.The process.The self-similar frog model SFM(d, p) has particles follow the same type of non-backtracking random walks as in nbFM(d, p) with some key amendments.The first modification is that we replace the single dormant particle at each site with independent Poi(1)-distributed numbers of particles.When an active particle visits a site with dormant particles, all dormant particles there become active.The additional modification is that particles moving away from the root are killed upon visiting a vertex that has already been visited.If multiple active particles attempt to move away from the root to the same unvisited vertex, then one is chosen to continue its path and the others are killed.Let V SFM(d,p) denote the total number of root visits in SFM(d, p).Notice that SFM(d, p) is defined with a Poisson distributed initial configuration of dormant particles.We use a result from [JJ18] to show that this can be compared to prove recurrence of FM(d, p).2.2.The operator.Given a probability measure π on the nonnegative integers, we define Aπ to be the self-similar frog model operator.It is obtained from the following auxiliary process.Consider a star graph with root ∅, central vertex ∅ , and leaves v 1 , . . ., v d (see Figure 1).There is a Poi(1) number of active particles at ∅ and a π-distributed number of active particles at v 1 .Independent π-distributed numbers of dormant particles are placed at v 2 , . . ., v d .
The active particles started at ∅ move to ∅ independently with probability p * and otherwise each moves to an independently and uniformly sampled vertex from v 1 , . . ., v d .Active particles at v i move to ∅ with probability 1, and then to either ∅ with probability p or otherwise to a uniformly sampled vertex among {v 1 , . . ., v d } \ {v i }.Whenever active particles encounter dormant particles, the dormant particles become active.When a particle moves to a leaf, it remains frozen there for all subsequent time steps.
Take Aπ to be the law for the total number of particles frozen at ∅ when the process fixates.Also, let Uπ be law for the total number of v 2 , . . ., v d that are ever visited by an active particle.We define U = U (d, p, λ) to be a random variable with distribution U Poi(λ).

Properties of A.
The following facts state that the law of V SFM(d,p) is a fixed point of A, that A is monotone, and that A Poi(λ) has a particularly nice representation.We also state [MSH03, Theorem 3.1 (b)] for comparing a Poisson random variable to one with a random parameter.We omit the proofs because they arise almost immediately from the construction and analogues have been observed in [HJJ16, JJ16, JMPR22].We will abuse notation and write AV SFM(d,p) and A Poi(λ) to represent the operator A applied to the associated probability measure.
We apply these facts to give a sufficient condition for recurrence.

Starting with V SFM(d,p)
Poi(λ 0 ) and iteratively applying Fact 6 and Fact 7 gives d, p) is recurrent, Lemma 5 ensures that so are nbFM(d, p) and FM(d, p).
Remark 11.The random variables U , Ũ , and U from Sections 3 and 5 are carefully balanced to satisfy (4).Expanding (4), we would like to show that For 1 + u ≥ 1/p, the summands are much smaller than e −λ .However, when 1 + u < 1/p, we need good bounds on the probability coefficients P(U = u) to make up for the e − p(1+u)λ terms being too large on their own.The balancing act is modifying U to obtain a smaller random variable with a tractable distribution that does not sacrifice too much precision.

Proof of Theorem 1
Fix p = 5/17 so that p * = (5d − 5)/(12d − 5) and p = 5/12.Note that p * is easily seen to be increasing in d.By taking d = 3 and d = ∞ we have 10/31 ≤ p * ≤ 5/12 for all d ≥ 3. (5) By Proposition 10, it suffices to find a random variable U U (d, p, λ) and > 0 so that (4) holds for all d ≥ 3. We define U to be the number of activated leaves in the following modified auxiliary process.First, we reduce to a star graph with only three leaves v 1 , v 2 , v 3 .Second, the Poi(1) active particles at ∅ move to ∅ with probability 10/31 ≤ p * , away from ∅ to a uniformly sampled leaf with probability 7/12 ≤ 1 − p * , and otherwise are immediately killed.Besides these changes, the process evolves in the same manner as the auxiliary process and runs until fixation.Then U ∈ {0, 1, 2} is how many of {v 2 , v 3 } are eventually visited by an active particle.
Proof.We first consider an intermediate process in which the active particles started at ∅ independently move away from ∅ to a uniformly sampled vertex from v 1 , ..., v d with probability 7/12 rather than 1 − p * .Let U d ∈ {0, 1, . . ., d − 1} be the number of activated leaves with this modification.Since 7/12 < 1 − p * , less particles are being sent to the leaves and thus U d U (d, 5/17, λ).Now the probabilities that active particles move towards the leaves are the same in both processes defining U d and U .From here, it is straightforward to couple the two random variables so that U U d for all d ≥ 3. It basically amounts to showing that a coupon collecting process with d versus 2 coupons has (stochastically) more unique coupons discovered after sampling the same number of coupons in each process.Thus, U U (d, 5/17, λ).
Proof of Theorem 1. Fix p = 5/17.We will show that (4) holds for all λ ≥ 0. By Lemma 12, and taking λ 0 = 0 in Proposition 10, we then have FM(d, 5/17) is recurrent for all d ≥ 3. Let U be the number of leaves from {v 1 , v 2 , v 3 } that are visited by an active particle in the modified auxiliary process with π = Poi(λ) for λ ≥ 0. Poisson thinning allows us to explicitly compute the distribution of U .Each of the Poi(1) active particles initially at ∅ visits one of the vertices {v 2 , v 3 } with probability 7 12 • 2 3 , and each of the Poi(λ) active particles initially at v 1 move to ∅ with probability 1, and then visits one of the vertices {v 2 , v 3 } with probability 7/12.Therefore, the probability that none of the vertices {v 2 , v 3 } is activated is the probability that a Poi( 7 12 2 3 + 7 12 λ) distributed random variable is equal to 0, giving rise to P(U = 0).One can apply the same argument to compute P(U = 1).The only difference is that if one of the vertices {v 2 , v 3 } is visited, particles initially there will be activated and can possibly visit the remaining unvisited vertex.
As a result of these calculations, we obtain x 6 e 10/31 + 1 e 397/558 , one can verify that g(e −λ/24 ) = f (λ).So, (7) is equivalent to the statement g(x) ≤ e − for some > 0 and all x ∈ [0, 1] (8) To prove (8) we first compute the derivative: [g ] = 6 from Wolfram Mathematica applies Sturm's theorem to rigorously find that g has exactly 6 roots in [0, 1].Since x = 0 is a root of order 5, g has exactly one root in (0, 1].Call it r 0 .Elementary calculus shows that g(r 0 ) is the global maximum on [0, 1].As g is an explicit polynomial, mathematical software can rigorously estimate both r 0 and g(r 0 ) to arbitrary precision.Doing so gives g(r 0 ) ≤ .9963< e −.003 .Thus, we may take = .003in (8).This gives (7) for the same which implies (4).As discussed at the onset of the proof, we have satisfied the hypotheses of Proposition 10.So, FM(d, 5/17) is recurrent for all d ≥ 3.
Assuming the reader has familiarity with the proof of Theorem 1, we now prove the generalization in Theorem 2. As in the proof of Theorem 1, we will show that (4) holds for all λ ≥ 0. We define Ũ := Ũ (d, p, λ) to be the number of activated leaves in the following modified auxiliary process.First, we reduce to a star graph with only four leaves ṽ1 , ṽ2 , ṽ3 , ṽ4 .Second, the Poi(1) active particles at ∅ move to ∅ with probability p = 81/265 ≤ p * , away from ∅ to a uniformly sampled leaf with probability ρ = 46/73 ≤ 1 − p * , and otherwise are immediately killed.Besides these changes, the process evolves in the same manner as the auxiliary process and runs until fixation.The random variable Ũ ∈ {0, 1, 2, 3} is how many of {ṽ 2 , ṽ3 , ṽ4 } are ever visited by an active particle.Following a similar argument as Lemma 12, we have Ũ U (d, 27/100, λ) for all d ≥ 4.
Using Poisson thinning, we can compute the distribution of Ũ : In words: { Ũ = 0} has no frogs from ṽ1 move to Ṽ := {ṽ 2 , ṽ3 , ṽ4 }; { Ũ = 1} has one vertex from Ṽ become activated (3 choices) and the other two fail to activate; and { Ũ = 2} has either two vertices from Ṽ initially activate (3 choices) and the third fail to activate, or one vertex from Ṽ initially activate (3 choices) and that activates exactly one more (2 choices), which then fails to activate the remaining vertex.
As in the proof of Theorem 1, we can write It suffices to prove that f (λ) ≤ e − for some > 0 and all λ ≥ 0. After the change of variables λ → −219 log x, this is equivalent to proving that g(x) := f (−219 log x) ≤ e − for some > 0 and all x ∈ [0, 1].The choice 219 was made from inspecting the expansion of f (computed with Mathematica) to find the least common denominator of the fractional exponents involving x.We check that CountRoots [0,1] [g ] = 70.Since g has a root of multiplicity 69 at x = 0, elementary calculus can be used to show that g has a global maximum in [0, 1] at x0 ≈ 0.992241 of g(x 0 ) ≈ 0.998772 < e −.0011 .These approximations are within 10 −7 of the true values, so we may take = 0.001 and complete the argument as in the proof of Theorem 1.
Remark 13.We describe how to generalize Theorem 2 to obtain a bound on sup d≥m p d .In principle, as m increases, this should give bounds closer and closer to 1/6 in agreement with Theorem 3. We did not try to go beyond m = 4, but this will likely become computationally infeasible at m ≈ 10.First we replace the d leaves with m leaves and construct Ũm ∈ {0, 1, . . ., m − 1} that is stochastically smaller than U (d, p, λ).This is accomplished by using the drift pm = p * (m, p) ≤ p * (d, p) towards ∅ in the auxiliary process and the drift ρ m = 1 − p ≤ 1 − p * (d, p) away from ∅ to a uniformly sampled child vertex from v 1 , ..., v m .We then need to compute the distribution of Ũm exactly.This is theoretically possible for any m, but becomes more and more complex as m grows.One then constructs a function fm (λ) = m−1 u=0 e − pm e (1−ρ(1+u))λ) P( Ũm = u) as at (9).One can plot fm using mathematical software to approximate small value of p for which fm (λ) < 1 for all λ ≥ 0, then use our approach that employs CountRoots to show that the transformation gm (x) < 1.

Non-backtracking branching random walk
In this section, we construct and deduce some properties of various spatially homogeneous multitype branching random walks that relate back to the frog model.4.1.Construction.The process starts with a configuration of particles on Z + at time 0. Each particle comes with a type i ∈ {1, 2, ..., k}.At discrete time steps, each particle independently gives birth to a random number of particles according to an offspring distribution that only depends on its type.The parent particle dies immediately after.Each newborn particle independently moves according to some displacement distribution that only depends on the particle's type.Particles that reach 0 are stopped there instantaneously and stay there forever without producing any offspring.
For i, j = 1, ..., k, let r ij be the expected number of offspring of Type-j produced by one Type-i particle and R = (r ij ) i,j=1,...,k be the mean matrix of the offspring distributions.For a particle of Type-i at site x ∈ Z ≥0 , we let p i x,y = P(Type i born at x moves to y).We now give a formal definition of the non-backtracking p-biased branching random walk on the nonnegative integers which we denote by nbBRW(p).Suppose a given particle is at x ≥ 1. Particles die immediately after producing offspring in the following manner: Type-1: Correspond to active particles that have yet to start moving away from the root.Each such particle produces either one Type-1 offspring with probability p, or one Type-2 offspring plus a Poi(1)-distributed number of Type-3 offspring with probability 1 − p.
Type-2: Correspond to active particles that have began to move away from the root.
Each such particle produces one Type-2 offspring and a Poi(1)-distributed number of Type-3 offspring with probability 1. Type-3: Auxiliary Type-1 particle, have the same offspring distribution as Type-1 particles, but different displacement distribution.
After producing offspring, each newly generated particle, independently of everything else, displaces from x according to the following transition probabilities: In words, Type-1 particles always move one step left, and Type-2 and Type-3 particles move one step right.We stop any particles that reach 0. Some quick remarks: • Type-2 particles correspond to non-backtracking active frogs that have turned away from the root and will continue moving away for all steps.
Type-1 and Type-3 particles correspond to non-backtracking active frogs that may still jump towards the root.We need two different particle types so that the displacements are independent of the manner in which particles are born.• There is no dependence on d in the definition of nbBRW(p).Since p * → p when d → ∞, one can view nbBRW(p) as the intuitive limiting version of nbFM(d, p).
Let V nbBRW(p) be the total number of particles that are killed at the origin.We say that nbBRW(p) is recurrent if P(V nbBRW(p) = ∞) = 1 and otherwise transient.We adapt ideas from [CMP98,MMP01] to find the criteria of recurrence and transience.
Let us first introduce some additional notation from [MMP01].For i = 1, 2, 3, we denote by N i (t) the number of Type-i particles at time t and {X i k (t) : k = 1, ..., N i (t)} the set of positions of Type-i particles at time t.Then the configuration at time t is the multiset The configuration of Type-i particles at time t is the multiset We denote p i = p i x,x−1 and q i = p i x,x+1 .Then we have p 1 = q 2 = q 3 = 1 and p 2 = p 3 = q 1 = 0.
Let M be the collection of initial configurations that consist of a finite number of particles distributed on Z ≥0 .Since for any i, j = 1, 2, 3, Type-i particles can be generated by a Type-j particle in finite steps with positive probability, we have either for all ω ∈ M. We will omit the initial configuration when we only care about the finiteness of E[V nbBRW(p) ] rather than its precise value.

4.2.
Transience and recurrence criteria.The following lemma is a combination of [MMP01, Theorem 4 and Theorem 7].It gives both necessary and sufficient conditions for nbBRW(p) to have a finite expected number of particles hitting the origin.The proof is a non trivial application of [MMP01, Theorem 4 and Theorem 7], because (a) nbBRW(p) does not satisfy all of the hypotheses used in [MMP01], and (b) the definitions of recurrence and transience in [MMP01] are different from our definitions.However, the proof ideas can be adapted to our case.We also note that results similar to Lemma 14 are present under other settings.A more general from of (10) appeared first in [KKS94] as a classification of one-dimensional branching random walk, then in [MV97] as a qualitative characterization of recurrence and transience for branching Markov chains, and also in [CMP98] under the setting of one-dimensional branching random walk in a random environment.The proof ideas are in the same vein.
Proof.If (10) holds, define The process {Q(t)} ∞ t=0 is a non-negative supermartingale.Indeed, let F t be the σ-field generated by nbBRW(p) up to time t.By the branching property and (10), we have By the supermartingale convergence theorem, there exists a random variable since only Type-1 particles can hit the origin.As a result, On the other hand, if to be the expected total number of visits to 0 conditional on the initial configuration starting with a single particle of type i at x.Note that f i (1) > 0 for i = 1, 2, 3.By the first step analysis, we have for i = 1, 2, 3 and x ∈ Z ≥1 , (12) r ij (p j f j (x − 1) + q j f j (x + 1)) .
For x ≥ 1, consider nbBRW(p) started from a single particle at x + 1 with Type-i.We can construct a modified process in which particles that reach the site 1 are stopped.Let V 1 be the number of particles that reach 1 and are stopped.For each lineage, only Type-1 particles can reach 1 for the first time.Note that V 1 has the same distribution as V nbBRW(p) under the process started from one Type-i particle at x. Therefore, V 1 < ∞ almost surely.Furthermore, because each Type-1 particle that reaches 1 behaves afterwards like another nbBRW(p) started from a single particle at 1 with Type-1, the number of particles stopped at 0 conditioned on V 1 is the same as the distribution of the sum of V 1 independent random variables, each with the same distribution as V nbBRW(p) under the process started from one Type-1 particle at 1.We therefore have for i = 1, 2, 3 By induction, we get for i = 1, 2, 3 and Plugging ( 13) into (12), by choosing µ = f 1 (1) and α i = f i (1) for i = 1, 2, 3, equation (10) holds with equality and the lemma follows.
Lemma 15.Suppose the initial configuration is finite and contains at least one Type-2 particle not at 0. Then nbBRW(p) is transient if and only if p ≤ 1/6.
Proof.Lemma 14 gives a criteria for proving that nbBRW(p) is transient.Namely, it is sufficient to prove that, when p ≤ 1/6, there exist θ ∈ R and α 1 , α 2 , α 3 > 0 such that equation (10) holds with equality for i = 1, 2, 3. Given θ ∈ R, define the weight matrix Equation (10) would hold with equality if we can show that 1 is the eigenvalue of Φ(θ) and there exists an eigenvector associated to 1 with all of its elements positive.The eigenvalues of Φ(θ) are with associated eigenvectors Solving β ± = 1 for θ gives the same solutions for both eigenvalues.In particular, For p ∈ (0, 1/5], the quadratic 1 − 6p + 5p 2 is non-negative and there is a solution.This implies that for p ∈ (0, 1/5], equivalently p ∈ (0, 1/6], there exists θ 0 with 1 the eigenvalue of Φ(θ 0 ).Furthermore, it can be easily shown that v + is an eigenvector of β + = 1 and every element of v + is positive.Equation (10) follows with equality and thus nbBRW(p) is transient for any p ≤ 1/6.On the other hand, if 1/6 < p < 1/2, then p ∈ (1/5, 1).Since the quadratic 1 − 6p + 5p 2 is negative for p ∈ (1/5, 1/2), there are no solutions to (10) in which equality holds.We know from Lemma 14 that E[V nbBRW(p) ] = ∞ for any finite initial configuration with not all particles at 0. It follows from the proofs of [MMP01, Theorem 9] and [CMP98, Theorem 4.3] that since nbBRW(p) is homogeneous in the sense that offspring distributions and transition probabilities do not depend on the location of particles, P(V nbBRW(p) ≥ 1) = 1 if the process starts from a finite number of particles not all located at 0. Moreover, if the initial configuration contains at least one Type-2 particle not at 0, then nbBRW(p) has Type-2 particles survive forever.Together with the Markovian property of nbBRW(p), we conclude that the origin is visited infinitely often almost surely.Therefore, P(V nbBRW(p) = ∞) = 1 and nbBRW(p) is recurrent.
Lastly, it is necessary for our arguments to deduce transience of a reflected version of nbBRW(p).Let reflected non-backtracking branching random walk rnbBRW(p) be the variant in which any particle that moves to 0, instead of being stopped, converts to a Type-2 particle that continues producing offspring.In rnbBRW(p), particles reflect at the origin.
Proof.Let V rnbBRW(p) be the number of times that particles hit the origin.Let A i,x denote the event ω(0) = ω i (0) = {x}.It is sufficient to prove that if p ≤ 1/6, then for all x ∈ Z ≥0 and i = 1, 2, 3 If x = 0, then equation ( 14) is obvious.Suppose rnbBRW(p) starts with one Type-i particle at site x ∈ Z >0 .If a particle hits the origin at time t, we can trace its past trajectory and count the total number of times this particle has hit the origin until time t.Call a visit to the origin an nth visit if the visiting particle has visited the origin exactly n − 1 times in its past trajectory.Let V rnbBRW(p) (n) be the total number of nth visits.We can decompose the expectation in ( 14) as For each n ≥ 1, we consider a modified process rnbBRW n (p) in which all particles are killed immediately after an nth visit.Note that for each n ≥ 1, the number of particles killed at the origin in rnbBRW n (p) is indeed V rnbBRW(p) (n).Furthermore, only Type-1 particles can visit the origin.In the original process RN BBRW n (p), the Type-1 particle is not killed after the visit.Instead, it will convert to a Type-2 particle and generate one Type-2 particle and Poi(1) Type-3 particle at site 1 in the next step.When n = 1, the modified process rnbBRW 1 (p) is identical to nbBRW(p).
4.3.Main ingredients in the proof Theorem 4. In Lemma 17, we show that transience of rnbBRW(p) when p ≤ 1/6 implies transience of nbFM(d, p).In Lemma 18, we use recurrence of nbBRW(p) when p > 1/6 to show that V SFM(d,p) dominates a Poisson random variable whose parameter diverges with d.
In regards to Lemma 17, intuitively nbBRW(p) ought to have more visits to 0 than nbFM(d, p) to the root.The reasons are (a) particles in nbBRW(p) have a slightly stronger drift towards 0 (because p * < p for p < 1/2), and (b) particles moving away from the root in nbBRW(p) always "activate" an additional Poi(1)number of particles.As discussed in the introduction, it is generally not known how to couple models with different drifts.Fortunately, (b) is enough to overcome these complications.
Overcoming the complications has the cost of a more involved coupling than might on the surface seem necessary.For example, we need to work with reflected branching random walk rnbBRW(p).Otherwise, in the killed-at-0 version (nbBRW(p)) the stronger drift might cause some particles to reach 0 and be killed, which hurts total progeny.We also introduce a family of (reflected) branching processes whose particle displacements depend on d.These are nice intermediaries that couple more cleanly with rnbBRW(p) and nbFM(d, p).
Proof.For this proof we will view nbBRW(p) as the process in which newly generated particles iteratively jump towards 0 with probability p, but once they turn away, continue to jump away from 0 for all subsequent steps.Each jump away from 0 produces an independent Poi(1)-distributed number of particles at the site jumped to.From this point of view, particles initially jump towards the root, eventually turn away, and then produce particles at each site they jump to thereafter.
Let rnbBRW(p), as introduced in Lemma 16, be the reflected modification.Any particle that visits 0, will on the next step jump to 1 and produce an additional Poi(1)-distributed number of particles there.Lastly, we define rnbBRW(d, p) to be the modification of rnbBRW(p) in which newly generated particles jump left on their first step with probability p * rather than p. Subsequent steps are to the left with probability p, as usual.
These transition probabilities are chosen so that rnbBRW(d, p) stochastically dominates nbFM(d, p).Namely, any active frog in nbFM(d, p) can be coupled with a unique particle in rnbBRW(d, p) whose position is equal to the active frog's displacement from the root of T d .The coupling is intuitive and works because (i) we may view rnbBRW(d, p) as the variant of nbFM(d, p) in which every jump away from the root activates new particles, and (ii) the transition probabilities towards and away from the root and 0 are the same for all steps of a particle's life in both models.This coupling ensures that transience of rnbBRW(d, p) implies transience of nbFM(d, p).
Since Lemma 16 gives transience of rnbBRW(p) whenever p ≤ 1/6, it suffices to prove that transience of rnbBRW(p) implies transience of rnbBRW(d, p).To this end, we may couple the initial particle at 0 in both rnbBRW(d, p) and rnbBRW(p) to introduce the same number of particles at each step away from the root.Since p * < p, any subsequently introduced particle in rnbBRW(d, p) can be coupled with a unique particle in rnbBRW(p), so that the particle in rnbBRW(p) moves at least at close to 0 before turning away.From there, we may couple the number of particles the two particles generate on their path to ∞ to be the same at each jump.This ensures that each particle in rnbBRW(d, p) corresponds to a unique particle in rnbBRW(p) that starts at least as close to 0 and moves at least as close to 0 as its counterpart.Thus, there are stochastically fewer total visits to 0 in rnbBRW(d, p).
Proof.Note that p * → p as d → ∞, and the probability of an active particle moving to an unvisited site within the first t time steps converges to 1 as d → ∞.Thus, the probability that SFM(d, p) and nbBRW(p) couple to have the exact same behavior for the first t time steps converges to 1 as d → ∞ for any fixed t ≥ 0. By Lemma 15, the probability of no root visits in nbBRW(d, p) goes to 0 as t → ∞.Letting V SFM(d,p) (t) be the number of root visits in SFM(d, p) up to time t, we then have P(V SFM(d,p) (t) = 0) → 0 as d → ∞.Poisson thinning ensures that the number of root visits in SFM(d, p) has a Poisson distribution with random mean.By Fact 9, V SFM(d,p) Poi(λ d ) for some sequence λ d → ∞.

Proof of Theorem 4
We use the criteria from Section 3 to prove that the limiting critical drift for nbFM(d, p) is the same as that in nbBRW(p).The basic idea is to construct a random variable U U from Section 2.2.We make it stochastically smaller by only allowing one leaf to be activated at a time.By taking d large, we are able to get sufficiently strong bounds on the probability that U takes a small value.
Before defining U , we describe an alternate way to sample U via an exploration process.Let A 1 be the set of leaves among v 2 , ..., v d that are visited by active particles started from ∅ and v 1 , and C 1 be the empty set.If A 1 is empty, then the process terminates.If not, then select a leaf v from A 1 and allow the activated particles there to move until reaching ∅ or a leaf.Let B 1 be the set of leaves among {v 2 , ..., v d } − {v} that are visited for the first time by particles from v. Set A 2 = A 1 ∪ B 1 − {v} and C 2 = C 1 ∪ {v}.Continue in this fashion to form A n+1 by removing a leaf v from A n , adding v to C n+1 , and adding the leaves visited for the first time by particles started from v to A n+1 .Once A N is empty (which occurs after at most d − 1 steps), we have C N is the set of leaves that are activated and so We define U U by modifying the exploration process.When B n is non-empty, instead of adding all of its leaves to A n to form A n+1 , we choose a single leaf from B n and add it to A n+1 .We ignore the visit of any other leaves in B n .In later rounds, all the leaves in B n except the one that was selected act as if the particles there are still dormant.Let U be the number of vertices activated in this modified process.
Since we are potentially ignoring visits to dormant particles we have U U.Moreover, for any 1 ≤ j ≤ 4 we have This is because in order to have U = j, the jth exploration must fail to visit any leaves with dormant particles.Since we are looking at the first j ≤ 5 steps, there are always at least d − 5 dormant leaves.Using Poisson thinning, the number of particles moving to leaves with dormant particles dominates a Poisson random variable with mean (1 − p)λ(d − 5)/(d − 1).So the probability of failure at the jth step, is bounded by the probability that this Poisson distribution is 0, which is (18).Lemma 19.Let U be as defined above.Given p > 1/6, there exists λ 0 ≥ 0 such that for all sufficiently large d and λ ≥ λ 0 E[e −p * − p(1+U )λ ] ≤ e −λ− 1 8 .(19) Proof.We first note that if p > 1/6, then p * ≥ 1/7 for all d ≥ 3. We expand the expectation at (19) to obtain Setting the e − 1 7 factor aside for a moment, we decompose the sum into three parts: Recall that for p > 1/6, p > 1/5 and thus 1 − 6p ≤ −1/5.Also, let δ > 0 be sufficiently small so that for d large enough Proof of Theorem 4. Lemma 17 implies that p d ≥ 1/6.For p > 1/6, Lemma 18 ensures that V SFM(d,p) Poi(λ 0 ) with λ 0 from the proof of Lemma 19 for all d large enough.It follows from Proposition 10 that SFM(d, p) is recurrent.By Lemma 5, nbFM(d, p) is recurrent.This gives Theorem 4.
Proof of Theorem 3. The result follows from Theorem 4 and Lemma 5.

Numerical simulations
We describe some numerical simulations to estimate p 3 , p 3 , and p 4 .The plots in Figures 2, 3, 4 were created using SageMath [The23], extending the code used in [HJJ17].Our code and a readme file are posted to the arXiv ancillary files for this article.The approach mirrors that of [HJJ17] with the appropriate adaptations for working with FM(d, p) and nbFM(d, p).To keep this section self-contained, we briefly review the method.
An adaptation of FM(d, p) or nbFM(d, p) is to insert stunning fences at each depth .The role of these fences is to stun (i.e., temporarily freeze in place) frogs when they first reach level .Let A d,p ( ) (resp.A d,p ( )) be the number of frogs in the FM(d, p) model (resp.nbFM(d, p)) that reach level once all awake frogs have been stunned.As noted in [HJJ17], counting the number of root visits in direct

Theorem 2 .
sup d≥4 p d ≤ 27/100.The authors of [BFJ + 19] further conjectured that lim d→∞ p d = q * : Corollary 5] states that recurrence of a frog model with Poisson initial conditions implies recurrence of the same model with one particle per site.The result follows from this and the construction in [GTW22, Section 2].The construction explains how SFM(d, p) is a restriction of nbFM(d, p), which is a restriction of FM(d, p).

Figure 2 .
Figure 2. Simulated values of s 3,p ( ) for FM(3, p) for different values of p, for levels ≤ 15.The blue points represent simulations of 1000 trials, and orange points represent simulations of 500 trials.

Figure 3 .
Figure 3. Simulated values of s 3,p ( ) for nbFM(3, p) for different values of p, for levels ≤ 15.The blue points represent simulations of 1000 trials, and orange points represent simulations of 500 trials.

Figure 4 .
Figure 4. Simulated values of s 4,p ( ) for nbFM(4, p) for different values of p, for levels ≤ 15.The blue points represent simulations of 1000 trials, and orange points represent simulations of 500 trials.