Resurgent Asymptotics in Physics

Updated 20 November 2025

Resurgent asymptotics for physics is a rigorous framework that decodes nonperturbative effects hidden in divergent series using Borel-Laplace resummation and Écalle’s alien calculus.
It unifies exponential asymptotics, Stokes phenomena, and trans-series analysis to bridge gaps between weak and strong coupling regimes in quantum mechanics, field theory, and hydrodynamics.
Numerical techniques like Borel-Padé approximants and conformal mappings enable accurate analytic continuation and global reconstruction of physical observables.

Resurgent asymptotics for physics is the rigorous framework that reveals how divergent perturbative expansions encode, within their large-order data, a full tower of nonperturbative corrections and the analytic structure of physical observables. This theory systematically unifies the heuristic insights of exponential asymptotics (such as the factorial-over-power ansatz for late-term series coefficients), the Stokes phenomena associated with turning points or singularities, and the modern machinery of Borel-Laplace resummation and Écalle's alien calculus. Resurgent asymptotics underpins pivotal developments in quantum mechanics, quantum field theory, statistical mechanics, topological string theory, hydrodynamics, and beyond, enabling both analytic and numerical extrapolation between weak and strong coupling, high and low temperature, or different regimes separated by singularities in parameter space.

1. Trans-Series Structure and Resurgent Formalism

Physical observables in singularly perturbed systems—such as asymptotics of solutions to nonlinear ODE/PDEs, path integrals, or partition functions—are generally represented not by convergent series but by formal trans-series: $y(\epsilon) \sim \sum_{m=0}^\infty a_m\,\epsilon^m + \sum_{k=1}^\infty C_k\,e^{-A_k/\epsilon}\;\nabla_k(\epsilon),$ where $A_k$ are singulants (instanton actions), $\nabla_k(\epsilon)$ are fluctuation series (often with fractional/negative powers and logarithms), and $C_k$ are Stokes multipliers. This reflects the breakdown of naive perturbation theory at exponentially small scales—so-called "effects beyond all orders"—which include tunneling amplitudes, instantons, capillary ripples, or eigenvalue tunneling effects in matrix models (Crew et al., 2022, Dunne, 19 Nov 2025).

Resurgence theory, stemming from Écalle's classification of resurgent functions, elucidates how these trans-series sectors are interconnected via bridge equations and Stokes automorphisms. All nonperturbative sectors (encoded in exponentially suppressed corrections) are determined from the perturbative sector by the analytic structure of the Borel transform.

2. Borel Transform and Factorial-Over-Power Asymptotics

The core step in resurgent asymptotics is to take a divergent series $\sum_n a_n \epsilon^n$ with $a_n \sim S\,\Gamma(n+\beta)\,A^{-n-\beta}$ as $n\to\infty$ , and pass to the Borel transform: $\mathcal{B}[y](w) = \sum_{n\ge1} \frac{a_n}{(n-1)!}\,w^{n-1}.$ Singularities of $\mathcal{B}[y](w)$ determine the nature, location, and magnitude of nonperturbative effects. A singularity at $w=A$ of fractional type $(A-w)^{-\alpha}$ yields the late-term coefficient structure—i.e., the universality of the factorial-over-power ansatz is grounded in the local Borel-plane analysis (Crew et al., 2022, Mas, 2019, Vonk, 2015).

The Stokes multiplier $S$ can be read off from the local expansion at the singularity; its value controls both the amplitude of the corresponding nonperturbative effect and, via Borel-Laplace inversion, the jump across Stokes lines in the parameter plane.

3. Discontinuities, Stokes Phenomena, and Alien Calculus

When the Laplace integral for summing the Borel transform crosses a singular direction (Stokes line), the observable jumps by an amount determined by the residue or branch-cut structure at the relevant singularity: $\mathrm{Disc}_\theta\,y(\epsilon) = \sum_{k} S_k\,e^{-A_k/\epsilon}\,\epsilon^{-\beta_k}\sum_{m\ge0} b_{k,m}\epsilon^m.$ In Écalle's framework, this is formalized via "alien derivatives" $\Delta_{A_k}$ (encoding the variation upon analytic continuation around the Borel singularity) and the Stokes automorphism: $\mathfrak{S}_\theta = \exp\left(\sum_{\arg A_k = \theta} e^{-A_k/\epsilon}\Delta_{A_k}\right),$ which encapsulates the intricate pattern of (nonlinear) Stokes phenomena in nonlinear, multiparameter ODEs—these generate, for example, the connection formulae for Painlevé I and II transseries and encode the nontrivial analytic monodromy for global solutions (Dunne, 19 Nov 2025, Baldino et al., 2022, Cleri et al., 2020).

4. Methods for Resurgent Extrapolation and Numerical Summation

Practical reconstruction of resurgent functions from asymptotic data employs combinations of Borel transforms, Padé approximants, and conformal mappings:

The Borel-Padé method constructs a rational approximation to the truncated Borel transform, extending analytic continuation past the radius of convergence and mimicking branch cuts with interlaced pole-zero patterns.
Conformal maps, such as $p\mapsto z=(\sqrt{1+p/p_1}-1)/(\sqrt{1+p/p_1}+1)$ , uniformize the cut Borel plane to a unit disk, optimizing Padé convergence and allowing resolution of closely spaced singularities (Costin et al., 2019, Costin et al., 2020).
Laplace inversion is then performed (possibly after contour deformation to avoid non-analyticities), yielding highly accurate numerical extrapolations—enabling, for example, global analytic continuation of Painlevé transcendents, matrix model free energies, or solutions to hydrodynamic attractor equations (Basar et al., 2015, Couso-Santamaría et al., 2015).

5. Applications in Physics and Mathematics

Resurgent asymptotics has direct implications across diverse domains:

Quantum Mechanics and Quantum Field Theory: Energy splitting in double-well potentials, resummation of perturbative series including instanton corrections, structure of renormalons in Dyson-Schwinger equations, and the analytic continuation between strong and weak coupling expansions (Borinsky et al., 2022, Mas, 2019).
Hydrodynamics and Attractor Solutions: The resurgent structure of gradient expansions in conformal plasma dynamics, identifying nonhydrodynamic relaxation modes as trans-series sectors determined by the ODE's Borel singularities (Basar et al., 2015).
Statistical Mechanics and Critical Phenomena: Extrapolation from high-temperature to low-temperature regimes, resolution of Lee–Yang edge singularities and phase transitions via conformal–Padé–Borel techniques (Costin et al., 2019).
Enumerative Geometry and Topological Strings: Factorial large-genus/large-degree growth of Gromov-Witten invariants, reconstruction of enumerative B-model data from A-model perturbation theory, nonperturbative transseries descriptions in mirror symmetry, and identification of Stokes data with D-brane/instanton amplitudes (Couso-Santamaría et al., 2016, Vonk, 2015, Eynard et al., 2023).
Matrix Models and Gravity: The interpretation of transseries sectors in terms of eigenvalue tunneling (ZZ/FZZT branes) and the analytic origin of Weil–Petersson large-genus volume asymptotics (Eynard et al., 2023).

6. Pathological and Advanced Phenomena: Coalescing Singularities, Nested Layers, and Logarithmic Asymptotics

Resurgent analysis provides systematic tools for handling cases where:

Singularities in the Borel plane coalesce (leading to enhanced, non-power-law asymptotics and intricate Stokes structures).
Nested boundary layers induce multiple competing singulants, requiring matched asymptotic expansions in hierarchical Borel variables.
The late-term expansion includes logarithmic or even non-classical (e.g., Airy function) singularities, as observed in renormalon ODEs, WKB/Witten Laplacian problems, or fractional polylogarithms (Crew et al., 2022, Getmanenko, 2010, Broadhurst, 30 Sep 2025).

Hybrid Borel-Hankel approaches and hierarchical matching techniques systematically account for these complications, confirming the universality and flexibility of the resurgent methodology.

7. Universality, Computational Protocol, and Outlook

The resurgent asymptotic protocol can be summarized as:

Construct the asymptotic expansion and identify the apparent divergence.
Form the Borel transform and investigate analytic continuability in the Borel plane, locating singularities and determining local exponents.
Approximate and continue the Borel transform via Padé and conformal maps as needed.
Compute Stokes multipliers via residue or monodromy calculation (or, if necessary, via numerical/analytic continuation).
Reconstruct the full observable—including all nonperturbative, beyond-all-orders effects—via Laplace inversion and analytic continuation.
In the presence of multiple singularities or coalescing saddles, utilize nested Borel expansions and Van Dyke-style matching.

This process "bootstraps" physical observables from perturbative data, enabling transitions between regimes not accessible to direct expansion and providing a systematic foundation for the nonperturbative completion of physics and geometry (Crew et al., 2022, Costin et al., 2019, Costin et al., 2020).

Resurgent asymptotics is thus a unifying and computationally powerful theoretical structure that delivers both conceptual understanding and practical analytic tools for extracting nonperturbative content from divergent series in physics. Its algebraic and analytic connections (through Borel-Laplace theory, Padé and conformal acceleration, and Écalle's alien calculus) have broad applicability and deep implications for the analytic, geometric, and numerical structure of physical and mathematical models.