Dynamical Fisher–Hartwig Asymptotics

• Dynamical Fisher–Hartwig asymptotics are rigorous expansions that characterize the time-dependent behavior of Toeplitz and Hankel determinants with moving or merging singularities.
• They employ advanced methodologies including Riemann–Hilbert analysis, steepest descent techniques, and differential identities with Painlevé transcendent connections to derive explicit asymptotic terms.
• This framework has practical applications in modeling non-intersecting particle systems, log-correlated fields, and dynamic phase transitions in stochastic and physical systems.

Dynamical Fisher–Hartwig Asymptotics

Dynamical Fisher–Hartwig (FH) asymptotics describe the large-size (and, more generally, large parameter) behavior of determinantal structures—especially Toeplitz, Hankel, and related integrable kernels—when the symbols exhibit time- or parameter-dependent singularities (root- or jump-type), with particular attention to scenarios where the singularities move, merge, or vary in strength dynamically. These asymptotics rigorously quantify leading-order and correction terms in a variety of time- or space–time-dependent models at the intersection of random matrix theory, non-intersecting particle systems, stochastic processes (e.g., stationary Hermitian Ornstein–Uhlenbeck processes), and log-correlated Gaussian fields/Gaussian multiplicative chaos. The theory generalizes classical, static FH asymptotic expansions by incorporating evolution of the singular data and their impact on global observables, fluctuations, and phase transitions.

1. Core Structure of Dynamical Fisher–Hartwig Asymptotics

The classical Fisher–Hartwig asymptotics express the behavior of Toeplitz (and, more generally, Fredholm, Wiener–Hopf, Hankel, Muttalib–Borodin) determinants of the form

$$D_N[\phi] \sim E[\phi]\, N^{-\sum_{j}\beta_j^2} \prod_{j} \frac{G(1+\beta_j)^2}{G(1+2\beta_j)} (1+o(1)),$$

where $\phi$ is a symbol with singularities at prescribed points (of jump or root-type), the $\beta_j$ are parameters characterizing the singularities, and $G(z)$ is the Barnes G-function.

In the dynamical context, the symbol $\phi = \phi_{t}(z)$ becomes time- or space–time dependent, typically as a result of evolution under a stochastic process (e.g., eigenvalues of a Hermitian Ornstein–Uhlenbeck matrix process, or the motion of singularities in parameter space). This dynamical evolution induces time-dependent exponents $\beta_j(t)$ and requires a uniform, multi-regime analysis to describe the (generally non-trivial) crossover between different asymptotic behaviors, including merging and separation of singularities (Claeys et al., 2014).

The generic dynamical FH asymptotic expansion then takes the form

$$D_N(t) \sim E(t)\, N^{-\sum_{j}\beta_j(t)^2} \prod_{j} \frac{G(1+\beta_j(t))^2}{G(1+2\beta_j(t))} \exp\big(C(t) + o(1)\big)$$

with explicit $C(t)$ depending on both the “background” data (smooth part, external potential, equilibrium measure) and the dynamical parameters.

2. Methodologies: Riemann–Hilbert, Steepest Descent, and Differential Identities

The standard mechanism for deriving these asymptotics involves steepest descent analysis of an associated Riemann–Hilbert problem (RHP) for orthogonal polynomials or integrable kernels built from the time-dependent symbol. The analytic steps are:

• Model RH problems and local parametrices: Construction of local parametrices at the (possibly merging) singularities, often mapped to Painlevé V (or IV) junctions in the double–scaling regime (Claeys et al., 2014, Charlier et al., 2021).
• Small-norm analysis and uniformity with respect to parameters: The “small-norm” RHP analysis produces remainder or error matrices (e.g., $R(z)$) whose size can be made $O(1/N)$—uniformly with respect to both the spatial/time parameters and position of the singularities, provided certain separation constraints are met (Forkel, 2023).
• Differential identities and integration: The leading order and correction terms are often extracted via differential identities with respect to the FH parameters (e.g., $\beta_j$) and the dynamical parameter (e.g., time $t$), integrated using known boundary data (Deift et al., 2012, Charlier et al., 2021).
• Painlevé transcendent connection: In regimes where singularities merge, the transition kernel is governed by a solution to a Painlevé V equation (Claeys et al., 2014), with its small and large argument expansions giving the local and crossover asymptotics.

The multi-regime structure of the asymptotics is encoded in the way the local model RH problems and their matching to the global parametrix depend on the dynamical variation of the singularities.

3. Dynamical Merging, Moving, and Scaling of Singularities

A distinctive haLLMark of the “dynamical” FH regime is the presence of time-dependent (or otherwise parameter-evolving) singularity data:

• Merging singularities: When two singularities approach each other as some dynamical parameter $t\to 0$, the asymptotic regime transitions from two separate singularities to a merged one (Claeys et al., 2014, Forkel, 2023). The corresponding determinant expansion switches from a product of two FH structures to a new merged regime characterized by a Painlevé V transcendent, effectively interpolating between the two asymptotic domains.
• Moving discontinuities/jumps: In spatially extended or space–time models (e.g., in non-intersecting Brownian motions (Keles, 15 Aug 2025)), the locations and intensities of the FH singularities evolve due to the stochastic or deterministic process governing the underlying system.
• Dynamical scaling and double-scaling limits: In certain situations, the strength of the singularity (e.g., the magnitude of a jump parameter or the mass of a root-type singularity) scales with the system size or time. For example, in the thinned GUE (Charlier et al., 2017), the size of the jump decays exponentially with $n$, inducing a transition in the determinant asymptotics described by a critical “dynamical” parameter $\lambda_c(t)$.

These dynamical modifications require analyzing the interplay between the rate of singular evolution and the global size parameter ($n, N, t$), often resulting in new physical regimes or phase transitions in the observable quantities (e.g., freezing transitions in log-correlated fields (Claeys et al., 2014, Berestycki et al., 2017, Keles, 15 Aug 2025)).

4. Connections to Stochastic Processes and Physical Models

Dynamical FH asymptotics appear in a range of mathematical physics problems:

• Random matrix theory with time-dependent dynamics: Under stationary Hermitian Ornstein–Uhlenbeck evolution, the eigenvalues flow according to an SDE preserving their non-crossing order (Keles, 15 Aug 2025). Determinant asymptotics for observables involving space-time singularities yield time-evolving FH exponents, leading to time-dependent scaling and phases.
• Non-intersecting Brownian motions: The probability that $N$ Brownian paths do not intersect up to time $t$ can be expressed in determinant form, with the exponent $\gamma(t)$ in

$$\mathbb{P}(\text{non-intersection over } [0, t]) \sim F(t) N^{-\gamma(t)}$$

determined by the squared sum of time-evolving FH exponents (Keles, 15 Aug 2025).

• Gaussian multiplicative chaos (GMC) and log-correlated fields: Fluctuations of the log characteristic polynomial, after dynamical centering/scaling, converge to log-correlated Gaussian fields in space-time. Moments of the determinant—via FH asymptotics—exhibit the correct multifractal scaling for convergence to GMC measures on strips or cylinders (Berestycki et al., 2017, Keles, 15 Aug 2025).
• Bulk rigidity and extremal statistics: The leading order of the log-characteristic polynomial and the precise scaling of the top eigenvalues (optimal rigidity) are governed by the dynamical FH structure, with the exponents' time-evolution yielding the correct bulk fluctuations (Keles, 15 Aug 2025).

5. Physical and Mathematical Implications

Dynamical Fisher–Hartwig asymptotics have several concrete outcomes:

• Explicit time-dependent formulas: The presence of time-dependent (or space–time-dependent) FH exponents in the determinant expansion enables the calculation of transient and crossover behavior in systems out of equilibrium or under parameter quenching (Keles, 15 Aug 2025, Claeys et al., 2014).
• Phase transitions and freezing phenomena: The competition between different “branches” (FH “realizations” related to periodicity and the multivaluedness of log) leads to non-analyticities in observable cumulants, corresponding to phase transitions in the statistical field theory (e.g., freezing transitions as in log-correlated fields (Claeys et al., 2014)).
• Universality across ensembles: The dynamical FH machinery extends across orthogonal, symplectic, and unitary invariant ensembles (Forkel, 2023), as well as to planar determinantal point processes with generalized symbols (Charlier, 2021), highlighting universality in diverse contexts.
• Cross-fertilization with integrable systems and spectral theory: Techniques from Riemann–Hilbert theory, differential equations (Painlevé transcendents), and advanced spectral theory underlie these expansions, rendering the field a focal point for the intersection of integrable systems, random matrices, and mathematical physics.

A representative theorem capturing the paradigm (Keles, 15 Aug 2025):

Dynamical Fisher–Hartwig Asymptotics Theorem: Under stationary Hermitian Ornstein–Uhlenbeck dynamics, let the Toeplitz determinant symbol have singularities characterized by time-dependent exponents $\beta_j(t)$. As $N \to \infty$,

$$D_N(t) = \exp(C(t) + o(1))\, N^{-\sum_{j}\beta_j(t)^2} \prod_{j=1}^r \frac{G(1+\beta_j(t))^2}{G(1+2\beta_j(t))},$$

with $C(t)$ explicitly computable from the specifics of the Ornstein–Uhlenbeck process, and where this asymptotic governs both the determinant's time evolution and the probability of non-intersecting particle paths in associated stochastic systems.

6. Representative Applications and Future Directions

Area	Dynamical FH role	Outcome or observable
Time-dependent random matrices	Determinant asymptotics	GMC, eigenvalue rigidity
Non-intersecting particle paths	Probability decay rates	Exponents from FH parameters
Log-correlated field theory	Moment asymptotics	Multifractal spectrum of GMC
Planar determinantal processes	Two-dimensional circular jumps	Disk counting cumulants, CLTs
Painlevé transcendents/dynamical	Double-scaling singularity merging	Crossover between phases/regimes

The field is expected to further connect with:

• Liouville quantum gravity via GMC measures in dynamics (Keles, 15 Aug 2025).
• Rigorous universality results for time-dependent random environments and “quenched” disordered processes.
• Detailed analysis of phase transitions in the evolution of fluctuation statistics, including states of freezing as in the maximal value of the log-characteristic polynomial (Keles, 15 Aug 2025).

7. Summary

Dynamical Fisher–Hartwig asymptotics provide a rigorous framework for understanding the evolution of determinantal observables with singular data that depend on time or other parameters. By extending classic, static results into time-dependent (stochastic or deterministic) settings—via advanced Riemann–Hilbert analysis, differential identities, and connections to Painlevé transcendents—these formulations have unveiled new universality classes, phase transition mechanisms, and scaling results across random matrix theory, integrable probability, log-correlated Gaussian fields, and complex stochastic systems (Keles, 15 Aug 2025, Claeys et al., 2014, Forkel, 2023, Charlier, 2021, Berestycki et al., 2017).