Cartwright local matrices as link walks with potential

Definitions and mathematical statement

Lionel Levine asks whether a simplicial-complex Laplacian generates a stochastic process; Matt Baker makes the question concrete for Dustin Cartwright's model. Caroline Klivans suggests Russell Lyons's work, which Levine says is interesting but does not answer his intended question; Andrea Sportiello points to Cartwright's weighting rule. The sign convention matters: a conservative continuous-time generator on functions is a matrix $Q$ with nonnegative off-diagonal entries and $Q\mathbf1=0$, whereas a positive graph Laplacian usually generates heat through $-L$.

The aid supplies a local Cartwright interpretation. At a codimension-two face $q$, let $A$ be the adjacency matrix of its finite link, counting a loop twice on the diagonal, let $d_i=\sum_jA_{ij}$, and let $\alpha_i$ denote the corresponding tropical structure constants. In the conventions of the aid, \[ M_q=A-\operatorname{diag}(\alpha),\qquad Q_{\mathrm{link}}=A-\operatorname{diag}(d),\qquad M_q=Q_{\mathrm{link}}+\operatorname{diag}(d-\alpha). \] The source does not provide the full structure-constant conventions, so this identifies the stated local model rather than every possible higher Laplacian. Its precise conservation criterion is \[ M_q\text{ is a conservative Markov generator}\iff \alpha_i=d_i\text{ for every }i. \] A scalar normalization $e^{-ct}e^{tM_q}$ is a Markov semigroup exactly when $d_i-\alpha_i=c$ for all $i$. In general, if $X_t$ is the link walk with generator $Q_{\mathrm{link}}$ and $V_i=d_i-\alpha_i$, the local representation is \[ (e^{tM_q}f)(i)={\mathbb{E}}_i\!\left[\exp\!\left(\int_0^tV_{X_s}\,{\,\mathrm{d}} s\right)f(X_t)\right]. \] This is a Feynman--Kac weighted expectation, not necessarily a probability-preserving evolution.

For comparison, let $H=\partial_{k+1}\partial_{k+1}^{\mathsf T}+\partial_k^{\mathsf T}\partial_k$ be a finite unweighted simplicial Hodge Laplacian in an oriented-cell basis. The aid reports a conservative signed-state representation of $e^{-tH}$ rather than direct cell-to-cell Markov evolution. An explicit version uses $r_i=\sum_{j\ne i}|H_{ij}|$ and any $c\geq\max(0,\max_i(r_i-H_{ii}))$. On states $(i,\sigma)$ with $\sigma\in\{-1,1\}$, jump to $(j,-\operatorname{sgn}(H_{ij})\sigma)$ at rate $|H_{ij}|$ for nonzero off-diagonal entries, and flip the sign at $i$ at rate $(H_{ii}+c-r_i)/2$. Its conservative chain $(I_t,\sigma_t)$ satisfies \[ (e^{-tH}f)(i)=e^{ct}{\mathbb{E}}_{(i,+1)}[\sigma_t f(I_t)]. \] The signs and scalar weight are essential. For the filled triangle the full edge Hodge Laplacian is $3I$, illustrating why its heat semigroup need not itself be a conservative walk moving among edges. These local and signed representations are supplied partial answers, not an identification of every simplicial Laplacian with an ordinary random walk.

Short English statement

Ask for stochastic interpretations of higher Laplacians, particularly Cartwright's model. The supplied aid represents local Cartwright matrices as link walks with a potential and gives the exact conservation criterion; general Hodge heat requires a signed or weighted representation rather than direct probability evolution on cells.

Sources

• AIM workshop Generalizations of chip-firing and the critical group, problem 36, Lionel Levine with comments by Caroline Klivans, Andrea Sportiello, and Matt Baker: https://aimath.org/pastworkshops/chipfiringproblems.pdf.
• The source refers to Russell Lyons as [24] (spelled "Russel" in the extract) and Dustin Cartwright as [13]; full entries are absent. Levine explicitly says the suggested Lyons work is not what he seeks.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001001, exact source code catalogue entry; canonical location aim-combinatorics-notes.json notes[125]. Attached attempt 1, partial result, problem status at run partially solved, rigor proved, low novelty confidence; machine-generated, not peer reviewed, requiring independent verification before citation.