Open research questions in Stochastic processes and statistical mechanics

What the literature leaves open

• The paper identifies a gap in the understanding of the ALE model and its relationship to the LPM. The authors note that the ALE model introduces an extra regularisation factor to deal with singularities in Loewner's equation.

  Tip growth in a strongly concentrated conformal growth model follows local geodesics · 2026 · DOI

• To prove the conjecture. To study the SDEP in more detail. To apply the results to real-world systems.

  Emergent hydrodynamics in an exclusion process with long-range interactions · 2026 · DOI

• The hydrodynamics of the SDEP were not well understood. The study of the SDEP can provide insights into the behavior of other lattice gases. The Doob transform provides a new tool for studying the SDEP.

  Emergent hydrodynamics in an exclusion process with long-range interactions · 2026 · DOI

• The complexity of the Poisson-Dirichlet process. The difficulty of computing the Bayesian posterior mean of the entropy and the Gini indexes. The lack of a Central Limit Theorem for the martingale difference.

  Bayesian Estimators of Diversity Indexes on Exchangeable Random Partitions · 2026 · DOI

• The non-Markovian nature of the persistence problem. The lack of a general framework for understanding the persistence problem. The need for new techniques for studying the persistence problem.

  Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution · 2026 · DOI

• The proof requires a technical assumption on the environment. The paper needs to provide a new understanding of the localization transition of the directed polymer.

  Strong disorder and very strong disorder are equivalent for directed polymers · 2026 · DOI

• Future research can focus on finding the precise value of the constant C. Future research can explore the applications of the results in various domains.

  The asymptotic behavior of rarely visited edges of the simple random walk · 2026 · DOI

• The paper suggests further research on the analysis of dynamical systems. The paper proposes the study of representations of fundamental groups as a future research direction.

  Dynamics on Markov surfaces: classification of stationary measures · 2026 · DOI

• The paper identifies a gap in prior work on the classification of S-invariant probability measures on S(C). The paper notes that prior work has not provided a classification of S-invariant probability measures on S(C).

  Dynamics on Markov surfaces: classification of stationary measures · 2026 · DOI

• The need for estimates of the SFS in stationary branching populations. The gap in understanding the SFS under the infinitely-many-sites assumption.

  Site Frequency Spectrum in stationary branching populations · 2026 · DOI

• The paper identifies certain technical challenges. The paper discusses the limitations of the existing literature. The paper identifies the gap in the existing literature.

  Asymptotics of lowlying Dirichlet eigenvalues of Witten Laplacians on domains in pinned path groups · 2026 · DOI

• Future research should consider certain cases. Future research should use certain techniques to estimate the norms of the functions. Future research should apply the results to various fields.

  Asymptotics of lowlying Dirichlet eigenvalues of Witten Laplacians on domains in pinned path groups · 2026 · DOI

• Future research can focus on extending the results to more general models. Future research can focus on applying the results to real-world datasets. Future research can focus on developing new methods for testing for Benford behavior.

  Benford Behavior Resulting From Stick and Box Fragmentation Processes · 2026 · DOI

• The existing research on Benford behavior is limited to single-proportion stick fragmentation models. There is a lack of understanding of Benford behavior in multi-proportion stick fragmentation models and box fragmentation models.

  Benford Behavior Resulting From Stick and Box Fragmentation Processes · 2026 · DOI

• Further study of the phase transition for other classes of graphs is needed. The results may be extended to other models of random processes on graphs.

  Existence and sharpness of the phase transition for the frog model on transitive graphs · 2026 · DOI

• The existence and sharpness of the phase transition for the frog model were not fully understood. The paper fills this gap by establishing the existence and sharpness of the phase transition for certain classes of graphs.

  Existence and sharpness of the phase transition for the frog model on transitive graphs · 2026 · DOI

• The rigorous understanding of the behavior of stochastic chains of oscillators is still limited.

  Stochastic oscillators out of equilibrium: scaling limits and correlation estimates · 2026 · DOI

• It has remained an open problem to find a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP.

  Type $C$ multiline queues and the open-boundary TASEP · 2026

• The need for a model that can overcome heteroscedasticity in flow data - The need for a model that can provide superior goodness of fit over the traditional lognormal model

  Further Discussion on the Poisson Interaction Model · 1983 · DOI

• Future research could consider the case where the variables are not nonnegative. Future research could consider the use of other inequalities and generalized inverses of distribution functions.

  On divergence and convergence of sums of nonnegative random variables · 1983 · DOI

• The paper identifies a gap in the literature by showing that the result has not been shown to hold without independence. The paper notes that prior work has assumed independence, but this paper shows that the result holds without this assumption.

  On divergence and convergence of sums of nonnegative random variables · 1983 · DOI

• The asymptotic behavior of the number of rarely visited edges is not well understood. The constant C is not precisely known.

  The asymptotic behavior of rarely visited edges of the simple random walk · 2026 · DOI

• In this note, we obtain a set of new boundary operators, Y (x) and e(k)(x) in (1.8), for the Brownian loop soup on the upper half-plane. We show that e(2k)(x) can be interpreted as an operator that inserts k outer boundaries of Brownian loops at the point x on the real axis. The physical interpretation of Y (x) and e(2k−1)(x) remains unclear. 5.1 Correlation functions of e(2k)(x) Since we already understand the physical interpretation of e(2k)(x), it would be interesting to compute their correlation functions, which, in principle, can be expressed in terms of the µbub ℍ -measure, as was done for ⟨e(2k)(x1)Oβ(z2)⟩ in section 3. For the bulk counterparts E (k) 0 (z), which insert the outer boundaries of k Brownian loops at the point z, the authors of [1] have written down explicitly the correlation functions of E (k) 0 (z) in terms of the Brownian loop measure µℍ that appears in (1.1). We leave the issue of finding the relation between correlation functions of e(2k)(x) and µbub ℍ , as well as their expressions, for future work. 5.2 Towards a CFT description of the Brownian loop soup While we have made progress in listing the boundary local operators in the spectrum of the CFT associated to the Brownian loop soup, understanding the CFT spectrum amounts to knowing not only the list of local operators, but also the model’s symmetries and the domains of all physical parameters. The last two points have not yet been fully understood. Here, we discuss possible ways of moving beyond the current knowledge. • Physical domain of β. Any function satisfying the conformal Ward identities can be decomposed into Virasoro conformal blocks. Nevertheless, the ability to decompose a collection of functions into conformal blocks does not automatically guarantee that those functions lead to consistent OPEs and are the n-point functions of a full-fledged CFT. This observation is relevant for the layering operator. In [11], where the layering operator was introduced, it was shown that the n-point functions ⟨Oβ1(z1) . . . Oβn(zn)⟩ exist and are conformally covariant for all n and all choices of parameters β1, . . . , βn. However, on the full plane, those n-point functions are nonzero only if P i βi = 2πn, n ∈ ℤ, a condition reminiscent of the charge neutrality condition that guarantees that the n-point functions of vertex operators do not vanish. Moreover, from previous results of [12], we observe that some structure constants of correlation functions of Oβ(z) for β = πn, n ∈ ℤ, admit simple factorizations in terms of the OPE coefficients. The same also happens for the upper half-plane two-point function ⟨Oπ(z1)O−π(z2)⟩, in which – 13 – JHEP04(2026)100 case some structure constants factorize into a product of OPE coefficients from [12] and the coefficients in (2.15) from the one-point functions in (2.13). This may suggest that it is possible to construct a consistent CFT describing various observables of the Brownian loop soup whose spectrum contains only layering operators Oβ(z) with β = πn, n ∈ ℤ. We suspect that the layering operators corresponding to other values of β may not be part of the spectrum of a consistent CFT. • The model’s symmetries. The authors of [12] have shown that the spectrum of the putative CFT describing the Brownian loop soup contains infinitely many bulk primary operators, whose scaling dimensions are positive integers. From Noether’s theorem, the existence of a primary operator of scaling dimension 1 is associated to a continuous global symmetry. In the Brownian loop soup, it remains unclear what the corresponding global symmetry is. Furthermore, the existence of primary operators of higher integer dimensions could imply higher symmetries. For instance, the 3-state Potts model has a primary operator of scaling dimension 3 associated to the W3-symmetry algebra, the 𝔰𝔩3-generalization of the Virasoro algebra [24]. It also remains unclear whether the Brownian loop soup has any higher symmetry associated to the primary operators of higher integer dimensions. Perhaps, the first step to understand the model’s symmetries would be to study the correlation functions of the primary operator of dimension 1. In particular, based on [25, 26], we expect that the limit λ → 0 of the two-point function of that operator is related to the area density of the region surrounded by the outer boundary of a Brownian loop.

  Boundary operators in the Brownian loop soup · 2026 · DOI

• The correlation coefficient ⟨kℓ⟩ = 7/4 (equation A27) is computed exactly, but the paper does not provide a systematic analysis of higher-order cumulants or joint moment asymptotics beyond the first and second moments, or establish how these moments relate to tree fragmentation properties under vertex removal.

  Leaves of preferential attachment trees · 2026 · DOI

• The generating function approach in equations (A5)-(A9) successfully solves the recurrence for the joint distribution, but the paper does not discuss whether this PDE method generalizes to related problems such as multivariate leaf statistics or higher-order correlations between degree and leafdegree in modified tree growth models.

  Leaves of preferential attachment trees · 2026 · DOI

Most-cited papers in Stochastic processes and statistical mechanics

• Frequency Distributions of Stream Link Lengths · The Journal of Geology · 1969 · 56 citations
• Bounded Confidence Revisited: What We Overlooked, Underestimated, and Got Wrong · Journal of Artificial Societies and Social Simulation · 2023 · 12 citations
• Influentials, early adopters, or random targets? Optimal seeding strategies under vertical differentiations · Decision Support Systems · 2024 · 7 citations
• Uniqueness in percolation theory · Statistica Neerlandica · 1994 · 5 citations
• Ergodicity of the Martyna-Klein-Tuckerman Thermostat and the 2014 Ian Snook Prize · Computational Methods in Science and Technology · 2020 · 4 citations
• On a conjecture of Seneta on regular variation of truncated moments · Publications de l Institut Mathematique · 2021 · 3 citations
• Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain · Topological Methods in Nonlinear Analysis · 2024 · 2 citations
• On central limit theorems for branching processes with dependent immigration · Bulletin of Taras Shevchenko National University of Kyiv Series Physics and Mathematics · 2020 · 1 citations
• Boundary operators in the Brownian loop soup · Journal of High Energy Physics · 2026 · 0 citations
• Approximation of the invariant measure for the stochastic Allen–Cahn equation via an explicit fully discrete scheme · BIT Numerical Mathematics · 2026 · 0 citations

Most recent work

• Boundary operators in the Brownian loop soup · Journal of High Energy Physics · 2026
• Approximation of the invariant measure for the stochastic Allen–Cahn equation via an explicit fully discrete scheme · BIT Numerical Mathematics · 2026
• Leaves of preferential attachment trees · Journal of Physics A: Mathematical and Theoretical · 2026
• Decay of Connection Probability in High-Dimensional Continuum Percolation · Mathematical Physics, Analysis and Geometry · 2026
• Scaling limit for the cover time of the λ-biased random walk on a binary tree with λ<1 · Bernoulli · 2026
• On A Roll Again: Analysis of a Dice Removal Game · The PUMP Journal of Undergraduate Research · 2026
• Tip growth in a strongly concentrated conformal growth model follows local geodesics · Probability Theory and Related Fields · 2026
• Emergent hydrodynamics in an exclusion process with long-range interactions · SciPost Physics · 2026
• Mixed Order Phase Transition in a Locally Constrained Exclusion Process · Journal of Statistical Physics · 2026
• Uniqueness of Dirichlet forms for random point fields in the absence of tail triviality · Probability Theory and Related Fields · 2026

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