Operator-Level Distribution: A Unified Approach

Updated 14 June 2026

Operator-level distribution is a modeling framework that focuses on fundamental operators, enabling precise mapping and analysis in complex systems.
It supports fine-grained operator mapping on heterogeneous hardware, achieving significant speedup, energy reduction, and optimal resource allocation.
The approach underpins rigorous proofs in quantum field theory, statistical mechanics, and optimization in power systems, highlighting cross-domain universality.

Operator-level distribution refers to the explicit modeling, optimization, and structural analysis of systems at the granularity of fundamental operators—be they computational, physical, or algebraic—rather than at the coarser task, block, or model level. The term encompasses frameworks for efficiently mapping and executing fine-grained operators over heterogeneous platforms, precise statistical mechanics of operator-valued random processes, operator-based formulations in statistical physics, as well as operator-centric approaches in stochastic integration, optimization, and energy systems. In contemporary research, operator-level distribution provides the abstraction necessary for both exploiting hardware heterogeneity in AI/machine learning inference and deriving meaningful statistical, combinatorial, or physical quantities in systems where the operator structure is fundamental.

1. Operator-Level Mapping in Heterogeneous Systems

Operator-level distribution is a foundational abstraction for assignments of computational primitives to hardware resources in heterogeneous platforms. The BIDENT framework introduces a model-agnostic orchestration of deep learning inference operators (e.g., convolutions, attention, non-linearities) over CPUs, GPUs, and NPUs in edge System-on-Chips (SoCs), utilizing detailed operator-wise profiling for latency and energy (Kim et al., 3 Jun 2026). BIDENT formulates the mapping problem as a shortest-path search in a directed weighted graph $G=(V,E)$ , with nodes corresponding to individual operator-PU assignments and edge weights capturing execution and transfer costs:

For operator $o_i$ on PU $P_j$ , node $v_{i,j}$ is added.
Edges $(v_{i-1,k}\to v_{i,j})$ carry weights $L_{i,j} + T_{k\to j}$ , where $L_{i,j}$ is profiled latency and $T_{k\to j}$ is inter-PU transfer overhead.
The optimal mapping corresponds to the $s\to t$ shortest path minimizing total cost:

$P^* = \arg\min_{P:\;s\leadsto t}\;\sum_{e\in P} w(e)$

This scheme is extensible to support:

Intra-model parallelism: Phase-based partitioning and per-branch optimal assignment with contention-aware makespan aggregation.
Multi-model concurrent scheduling: Construction of a product graph state space to co-optimize operator execution across multiple inference streams.

Empirical evaluation on an Intel heterogeneous SoC demonstrates substantial performance and energy benefits: up to 1.60× speedup with parallelism and 3.42× geometric mean speedup in multi-model scheduling, with concurrent energy reduction of 48.2% (Kim et al., 3 Jun 2026).

2. Operator-Level Factorization in Quantum Field Theory

In high-energy physics, the operator-level distribution framework enables rigorous proofs of factorization theorems. For example, the Drell-Yan and SIDIS processes in QCD are shown to factorize at the operator level by carefully analyzing the cancellation of soft gluon exchanges and contour deformations due to unitarity and eikonal approximations (Zhou, 2013, Zhou, 2013). Key elements are:

PDF and FF definitions: Gauge-invariant operator-valued distributions expressed with explicit Wilson lines.
Unitarity-induced cancellations: Post-collision soft interactions vanish, enabling decoupling and factorized structure.

Operator-level approaches eliminate the need for diagrammatic combinatorics, yielding robust and minimal factorized cross-section representations in terms of operator expectation values.

3. Operator-Level Distributions in Random Matrix and Many-Body Physics

The concept of operator-level distribution is integral in advanced statistical mechanics and random matrix theory:

Operator spectrum distributions: The spectrum of operator-valued quantities, such as the Lyapunov operator constructed from out-of-time-ordered commutators, exhibits universal Wigner–Dyson statistics, bridging classical exponential instability and fully quantum chaos (Rozenbaum et al., 2018). At short times, the spectrum matches finite-time Lyapunov exponents, while at longer times, it exhibits level repulsion due to quantum interference.
Operator-size distributions: In models of quantum many-body dynamics (e.g., Brownian SYK), the operator-size distribution $o_i$ 0—the probability that an evolving Heisenberg operator has support size $o_i$ 1—obeys an exact master equation. Analytical solutions in large $o_i$ 2 yield universal $o_i$ 3 forms for the full size distribution, with precise control of global and sector-specific features (Xu, 2024, Tarakemeh et al., 20 Nov 2025).

Domain	Operator-object	Distribution concept
Heterogeneous inference (Kim et al., 3 Jun 2026)	DL graph operator	PU-mapping cost minimization
Quantum chaos (Rozenbaum et al., 2018)	Lyapunov operator spectrum	Wigner–Dyson level statistics
Many-body dynamics (Xu, 2024)	Operator size	Master equation for time-dependent $o_i$ 4

4. Operator-Level Distribution in Power Systems and Market Design

In energy systems, operator-level distribution describes optimization, market clearing, and coordination of distributed assets (DERs, loads, aggregators) subject to physical constraints:

Bilevel optimization for TSO-DSO coordination: Hierarchical models with Distribution System Operators (DSOs) as leaders performing operator-level scheduling of distributed resources (batteries, PV, P2P exchanges), while the Transmission System Operator (TSO) executes lower-level dispatch. Formulations exploit mixed-integer second-order cone programming (MISOCP) to preserve the integrity of discrete operational logic in DER/P2P control (García-Muñoz et al., 24 Mar 2026).
Market clearing and settlement: Operator-level market frameworks compute locational marginal prices (DLMPs) at distribution buses by solving convex welfare maximization subjected to all grid constraints (Parhizi et al., 2016). The dual variables of nodal energy balance equations yield DLMPs, which aggregate upstream transmission cost and local congestion effects:

$o_i$ 5

Distributed allocation via bi-level auction: Operator-level decomposition into network-constrained DSO allocations (upper level) and privacy-preserving aggregator micro-auctions (lower level), ensuring global feasibility, social welfare maximization, budget balance, and user privacy (Faqiry et al., 2017).

5. Mathematical Foundations of Operator-Level Probability Distributions

Several domains employ operator-level probability distributions either through operator-fixed-point characterizations or limiting object identification:

Operator-Dickman distributions: The multidimensional Dickman law and its operator-level generalization D(Q, ν) is defined as the unique fixed point of the stochastic affine equation $o_i$ $o_{i}$ 6 where $o_i$ $o_{i}$ 7 Uniform $o_i$ $o_{i}$ 8, $o_i$ $o_{i}$ 9 is a real $P_j$ $P_{j}$ 0 matrix, and $P_j$ $P_{j}$ 1 is a random jump measure (Kovtun et al., 13 Feb 2026). These measures are:
- Infinitely divisible, with explicit Lévy–Khintchine representation.
- Operator-selfdecomposable, with characteristic function
$P_j$ 2 - Universal in the sense of providing scaling limits of record-time distributions, Lévy process small-jump approximations, and infinite convolution closures.
Operator-level limiting processes in random matrix theory: Circular Jacobi and real-orthogonal β-ensembles are realized as spectral measures of Dirac-type operators. Operator-level convergence theorems ensure almost-sure strong resolvent convergence, providing unified access to point process limits, secular function convergence, and SDE-based counting function representations (Li et al., 2021). Similar techniques underlie the norm-resolvent convergence for hard-to-soft edge transitions in β-ensembles (Dumaz et al., 2020).

6. Extensions and Cross-Domain Universality

Operator-level distribution frameworks exhibit several universal signatures across application areas:

Unified operator-centric modeling: Operator-level abstraction captures the full system, maintaining structural invariants even in stochastic, heterogeneous, or adversarial regimes.
Computational tractability: Many operator-level assignment problems reduce to polynomial-time combinatorial optimization (shortest path, dynamic programming), even for large heterogeneous or high-dimensional systems (Kim et al., 3 Jun 2026, García-Muñoz et al., 24 Mar 2026).
Emergent universality: Universal distributional forms (e.g., $P_j$ 3 laws, Wigner–Dyson spacing) arise independently in quantum, statistical, and applied optimization contexts, reflecting deep operational symmetries and constraint structures (Xu, 2024, Rozenbaum et al., 2018).

Operator-level approaches eliminate heuristic or coarse-grained partitioning by working at the grain of structural, algebraic, or executable primitives, ensuring optimality, maximal parallelism, and fundamental insight into distributional characteristics throughout complex, multi-component systems.