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arxiv: 2606.27083 · v1 · pith:2XNJ63JBnew · submitted 2026-06-25 · 🧮 math.SP · math-ph· math.CA· math.MP

Discrete Space-Time Wave Kernels on Regular Trees

Amar Ba\v{s}i\'c , Lejla Smajlovi\'c , Zenan \v{S}abanac This is my paper

Pith reviewed 2026-06-26 01:31 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.CAmath.MP
keywords discrete wave equationregular treesgeneralized LaplacianI-Bessel functionsJ-Bessel functionswave kernelsspectral theory
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The pith

Explicit formulas express the two fundamental wave kernels on regular trees via discrete I-Bessel functions under a nonnegativity assumption.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit formulas for the two fundamental solutions of the discrete wave equation on a homogeneous (q+1)-regular tree equipped with a two-parameter generalized Laplacian. These formulas, expressed through discrete I-Bessel functions, produce convolution representations that solve the equation for arbitrary initial data. When the operator reaches the bottom of its spectrum, the kernels admit an alternative expression in discrete J-Bessel functions, establishing a discrete counterpart to the classical relation between I and J functions. The expressions consist of finite sums, ensuring that time-stepping remains exact and finite at every discrete time. Analytic and numerical asymptotics are provided for large distances, times, and tree branching.

Core claim

Under the natural nonnegativity assumption on this operator, we derive explicit formulas for the two fundamental wave kernels. The formulas are given in terms of discrete I-Bessel functions and yield convolution representations for solutions with general initial conditions. In the boundary case corresponding to the bottom of the spectrum, we obtain another explicit representation of the wave kernel in terms of discrete J-Bessel functions. This representation leads to a discrete analogue of the classical I↔J relation. The wave kernels are expressed as finite sums; hence, the propagation formulas remain finite for every discrete time.

What carries the argument

The two-parameter generalized Laplacian on the (q+1)-regular tree, with discrete I-Bessel functions providing the explicit finite-sum form of the wave kernels.

If this is right

  • Solutions with general initial conditions are obtained directly as convolutions against the kernels.
  • Propagation formulas stay finite sums at every discrete time step.
  • Asymptotic expansions hold for large radial distance, large time, and large tree degree.
  • A discrete version of the classical I to J Bessel relation appears at the spectral bottom.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite-sum property could support exact long-time evolution without truncation on finite subtrees.
  • The same Bessel-based approach might adapt to wave problems on other vertex-transitive graphs with similar radial symmetry.
  • The explicit kernels may connect to counting problems or generating functions on trees in combinatorics.

Load-bearing premise

The two-parameter generalized Laplacian must satisfy the natural nonnegativity assumption for the explicit kernel formulas to hold.

What would settle it

Numerical solution of the wave equation on a small regular tree with chosen parameters satisfying nonnegativity, compared term-by-term to the proposed I-Bessel formula, would falsify the claim if the two disagree at any time step.

Figures

Figures reproduced from arXiv: 2606.27083 by Amar Ba\v{s}i\'c, Lejla Smajlovi\'c, Zenan \v{S}abanac.

Figure 1
Figure 1. Figure 1: Signed logarithmic transform of W 3−2 √ 2,1 3 (r;t) for r = 1, 2, 5. Corollary 6.3. Let q ≥ 2. For a fixed radial distance r, the wave kernel W b(q+1−2 √q),b q+1 (r;t) oscillates in time t with exponentially growing amplitude. In other words, for θ ∈ (0, π) such that cos θ = (1 + 4b √q) −1/2 we have W b(q+1−2 √q),b q+1 (r;t) ∼ (−1)r q −r/2 √ πtsin θ Cb(r;t) (1 + 4b √ q) t 2 cos t + 1 2  θ − π 4  , as t… view at source ↗
Figure 2
Figure 2. Figure 2: Signed logarithmic transform of W 4−2 √ 3,1 4 (r;t) for r = 1, 2, 5. Each difference of J-Bessel functions in the above sum can be approximated using (6.1). More￾over, we note that the second line of (6.1) is equal to a quotient of two polynomials in the variable t, both of degree 4, which implies that their limit as t → ∞ is a nonzero constant. Therefore, for θ ∈ (0, π) such that cos θ = (1 + 4b √q) −1/2 … view at source ↗
Figure 3
Figure 3. Figure 3: Signed logarithmic transform of W 0, 1 3 (r;t) for r = 1, 2, 5. The growth in this case is significantly faster than in the boundary case a = b(q + 1 − 2 √q). That is a consequence of the fact that the magnitude of oscillations of W a, b q+1(r;t) for large t is approximately (1 +b(q + 1 + 2√q)−a) t/2 , which, for a = 0 and b = 1, equals (2 +q + 2√q) t/2 , a number that is significantly larger for q ≥ 2 tha… view at source ↗
Figure 4
Figure 4. Figure 4: Signed logarithmic transform of W 0, 1 4 (r;t) for r = 1, 2, 5. The largest eigenvalue for the normalized/probabilistic Laplacian is λmax = 1 + 2 √q q+1 , which is significantly smaller than q + 1 + 2√q, the largest eigenvalue of the combinatorial Laplacian on Tq+1. Hence, the oscillations of the wave kernel W 0,1/(q+1) q+1 associated with the probabilistic Laplacian exhibit smaller, yet still exponential,… view at source ↗
Figure 5
Figure 5. Figure 5: Signed logarithmic transform of W 0, 1 3 3 (r;t) for r = 1, 2, 5. When a and b are chosen so that the largest eigenvalue is close to 0 (yet positive, due to the bound (1.1)), the magnitude of the oscillations is smaller, as illustrated in Table A.3 in the Appendix and shown in [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Signed logarithmic transform of W 0, 1 4 4 (r;t) for r = 1, 2, 5. ● ● ● ● ● ● ● ● ● ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● r = 1 ▲ r = 2 ■ r = 5 200 400 600 800 1000 -30 -20 -10 0 10 20 30 t log |W| [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Signed logarithmic transform of W 1/100, 1/100 5 (r;t) for r = 1, 2, 5. is interesting to observe that, as q → ∞, the wave kernel W 1− 2 √q q+1 , 1 q+1 q+1 associated with the probabilistic Laplacian in the boundary case tends to zero (meaning that, as the edges in the graph become dense, the wave kernel tends to zero for fixed time and fixed radial distance). More precisely, we have the following corollar… view at source ↗
Figure 8
Figure 8. Figure 8: Asymptotic behavior of the wave kernel W 1− 2 √q q+1 , 1 q+1 q+1 (2;t) as a function of q for t = 30, 40, 50. 7. Fundamental solutions when ∆ a,b q+1 is not necessarily semipositive In this section, we study the situation in which the parameters a, b (b ̸= 0) are arbitrary, meaning that (1.1) is not assumed, hence the spectrum of the operator ∆ a,b q+1 may be negative. The results of Sections 2 and 3 remai… view at source ↗
Figure 9
Figure 9. Figure 9: Signed logarithmic transform of W 2,1/3 3 (r;t) for r = 1, 2, 5. References [1] L. Abadias, E. Alvarez, S. Díaz, Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation, J. Math. Anal. Appl. 507 (2022), no. 1, Article ID 125741, 23 p. [2] L. Abadias, J. González–Camus, S. Rueda, Time-step heat problem on the mesh: asymptotic behavior and deca… view at source ↗
read the original abstract

We study the forward discrete space-time wave equation on the homogeneous $(q+1)$-regular tree $T_{q+1}$ associated with a two-parameter generalized Laplacian. Under the natural nonnegativity assumption on this operator, we derive explicit formulas for the two fundamental wave kernels. The formulas are given in terms of discrete $I$-Bessel functions and yield convolution representations for solutions with general initial conditions. In the boundary case corresponding to the bottom of the spectrum, we obtain another explicit representation of the wave kernel in terms of discrete $J$-Bessel functions. This representation leads to a discrete analogue of the classical $I\!\leftrightarrow\!J$ relation. We also perform both analytic and numerical studies of the asymptotic behavior of the wave kernels, including large radial distance, large time, and large degree of the tree. An important feature of our analysis is that the wave kernels are expressed as finite sums; hence, the propagation formulas remain finite for every discrete time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper derives explicit finite-sum formulas for the two fundamental wave kernels associated to the discrete space-time wave equation on the (q+1)-regular tree T_{q+1} for a two-parameter generalized Laplacian. Under the stated nonnegativity assumption on the operator, the kernels are expressed via discrete I-Bessel functions and yield convolution representations for general initial data. In the boundary case at the bottom of the spectrum, an explicit J-Bessel representation is obtained, producing a discrete analogue of the classical I↔J identity. Analytic and numerical asymptotics are provided for large radial distance, large time, and large degree q, with emphasis on the fact that the finite-sum expressions keep the propagation formulas exact at every discrete time.

Significance. If the derivations hold, the explicit finite-sum kernels constitute a concrete advance for spectral theory on infinite regular graphs and discrete wave equations. The convolution representations and the discrete I-J identity are useful structural results, while the finite-sum property enables exact computation without truncation error at finite times. The combination of analytic asymptotics with numerical verification strengthens the contribution for applications in discrete PDEs and graph Laplacians.

minor comments (3)
  1. [§2] The precise definition of the two-parameter generalized Laplacian and the exact statement of the nonnegativity assumption should be recalled at the beginning of the derivation section (currently referenced only in the abstract) to make the domain of validity of the I-Bessel formulas immediately clear.
  2. [§3] Notation for the discrete I-Bessel and J-Bessel functions should be introduced with a short self-contained definition or reference to the precise recurrence they satisfy, rather than assuming familiarity from prior literature on discrete Bessel functions.
  3. [§5] In the asymptotic analysis, the transition between the large-time and large-distance regimes could be stated more explicitly; currently the error terms in the large-q expansion are described qualitatively but without an explicit remainder estimate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and detailed summary of our work, the assessment of its significance, and the recommendation of minor revision. No specific major comments appear in the report, so we have no individual points to address point-by-point. We remain ready to incorporate any minor editorial or clarification changes the editor or referee may suggest in a subsequent round.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives explicit finite-sum formulas for the wave kernels directly from the spectral properties of the two-parameter generalized Laplacian on the regular tree, under an explicitly stated nonnegativity assumption that defines the regime of validity. These formulas are expressed via discrete I-Bessel functions and yield convolution representations; the boundary-case J-Bessel form produces a discrete I↔J identity as a consequence. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the derivations are presented as analytic consequences of the operator and remain finite for each discrete time. The manuscript is self-contained against external benchmarks with independent analytic and numerical asymptotics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central derivations rest on a single domain assumption stated in the abstract; no free parameters or new postulated entities are introduced.

axioms (1)
  • domain assumption natural nonnegativity assumption on the two-parameter generalized Laplacian
    Explicitly required in the abstract for the derivation of the I-Bessel and J-Bessel kernel formulas to hold.

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Reference graph

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