Open research questions in Stochastic processes and statistical mechanics
133 gap statements mined from Stochastic processes and statistical mechanics papers in our 4.5M-paper local library, which holds 539 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The physical interpretation of Y (x) and e(2k−1)(x) remains unclear. 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.
Introducing a weak asymmetry or external noise is a natural route, yet it remains unclear which—if any—perturbations actually generate KPZ-type scaling. At the PDE level, a Hilbert-kernel variant of the KPZ equation has already been proposed earlier; clarifying its connection to the SDEP is an open problem.
Future research could build on the results of this paper to develop new models and algorithms for stochastic systems with conditional drift. The paper's findings could be applied to the study of self-interacting random processes and other fields such as physics and engineering.
Limit theorems for random walks with spatio-temporal drift · 2026Future research can apply the approach used in this paper to other models, such as CSS codes. The result can be used to study the mixing times of other quantum systems. Future research can aim to establish an MLSI for other models.
Modified logarithmic Sobolev inequalities for Abelian quantum double models · 2026The lack of MLSI results for Abelian quantum double models. The need for a technique to establish an MLSI for these models. The gap in prior work is the lack of results for these models.
Modified logarithmic Sobolev inequalities for Abelian quantum double models · 2026The lack of a Central Limit Theorem for the martingale difference.
The need for a deeper understanding of inhomogeneous random graphs and their asymptotics. The lack of advanced techniques for analyzing the connectivity and behavior of these graphs. The importance of establishing functional central limit theorems for microscopic and macroscopic functionals.
Functional central limit theorems for microscopic and macroscopic functionals of inhomogeneous random graphs · 2026 · DOIThe study uses a simplified model that may not capture all the complexities of urban growth. The dataset is limited to 1,567 cities worldwide, and may not be representative of all urban areas. The study focuses on building heights and does not account for other factors that may influence urban growth.
The paper focuses on a specific type of GWI process. The analysis is limited to the asymptotic behavior of the process. The paper does not provide a detailed analysis of the applications of the results.
Asymptotic behavior of some strongly critical decomposable 3-type Galton–Watson processes with immigration · 2026 · DOIFuture research should focus on extending the results to more general cases of GWI processes. The paper suggests that further analysis of the applications of the results is needed.
Asymptotic behavior of some strongly critical decomposable 3-type Galton–Watson processes with immigration · 2026 · DOIFuture research could extend the results to other areas of finance. Future research could explore the applicability of the results to other types of martingales.
Prior work has not provided a general representation of martingales with independent increments. The gap is that prior work has not provided a representation of martingales with independent increments as sums of Gaussian martingales.
The paper assumes that the birth rate is age-dependent and random. The authors do not require that b and η be independent. The results are limited to the case where the process starts with a single individual at time 0.
Functional law of large numbers and central limit theorem for Crump–Mode–Jagers branching processes · 2026 · DOIThe authors suggest that future research could focus on extending the results to more general processes. The paper could be used as a starting point for studying other types of branching processes.
Functional law of large numbers and central limit theorem for Crump–Mode–Jagers branching processes · 2026 · DOIFuture research should investigate the applications of germ order in other fields. Future research should develop new models and methods for the study of branching random walks and rumor processes.
The gap is the lack of understanding of the properties of germ order in multidimensional generating functions. The gap is the need for new conditions for survival or extinction of branching processes in varying environments.
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.
The existing literature does not provide a complete analysis of the conditional drift and its polynomial decay in time and growth with distance.
Limit theorems for random walks with spatio-temporal drift · 2026To establish a Central Limit Theorem for the martingale difference. To extend the results to other prior distributions.
To extend the model to account for other factors that may influence urban growth, such as demographic change and technological development. To apply the model to other urban areas and to test its generalizability. To use the model to inform decision-making on urban development and resource allocation.
To consider the ergodicity of the numerical invariant measure. To study the approximation error between numerical invariant measure and the original one.
Approximation of the invariant measure for the stochastic Allen–Cahn equation via an explicit fully discrete scheme · 2026 · DOIAnalysis of the joint degree-leafdegree distribution in other tree growth models. Application of the results to real-world networks. Study of the isotropic complete redirection model.
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.
Further study of the novel variant of the exclusion process with a next-nearest-neighbor constraint. Investigation of the system's behavior at densities above half filling.
The paper identifies a gap in the understanding of kinetically constrained lattice gases with next-nearest-neighbor constraints. The paper aims to fill this gap by introducing a novel variant of the exclusion process.
Most-cited papers in Stochastic processes and statistical mechanics
- Statistical Law of Stream Numbers · The Journal of Geology · 1966 · 693 citations
- Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions · Operations Research · 1984 · 429 citations
- Infinite Topologically Random Channel Networks · The Journal of Geology · 1967 · 391 citations
- Stationarity and the existence of moments of a family of GARCH processes · Journal of Econometrics · 2002 · 231 citations
- Stream Lengths and Basin Areas in Topologically Random Channel Networks · The Journal of Geology · 1969 · 222 citations
- Poisson Approximation and the Chen-Stein Method · Statistical Science · 1990 · 194 citations
- Properties of moments of a family of GARCH processes · Journal of Econometrics · 1999 · 169 citations
- Random Walk and the Theory of Brownian Motion · American Mathematical Monthly · 1947 · 161 citations
- Markov Random Fields and Gibbs Ensembles · American Mathematical Monthly · 1971 · 150 citations
- Kemeny's Constant and the Random Surfer · American Mathematical Monthly · 2002 · 147 citations
Most recent work
- Conservative Stochastic PDE and Fluctuations of the Symmetric Simple Exclusion Process · Communications in Mathematical Physics · 2026
- Hydrodynamic Limit for an Open Facilitated Exclusion Process with Slow and Fast Boundaries · Communications in Mathematical Physics · 2026
- Dynamic roughening of cities driven by multiplicative noise · Nature Communications · 2026
- Benford Behavior Resulting From Stick and Box Fragmentation Processes · Journal of Number Theory · 2026
- Random irregular histograms · Computational Statistics & Data Analysis · 2026
- Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes · Journal of Mathematical Analysis and Applications · 2026
- Mass functions and asymptotic behavior of caloric functions on affine buildings · Mathematische Annalen · 2026
- 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
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