Geometric Criticality

• Geometric criticality is a phenomenon where critical behavior emerges from changes in structures like filtrations, curvature, or network topology.
• It serves both as a mechanism and diagnostic tool in phase transitions, evidenced by p-adic models, quantum geometric tensors, and information geometry.
• Research in this area offers practical insights into percolative networks, black-hole observables, and anomalies in thermodynamic and structural systems.

Searching arXiv for papers on “geometric criticality” and closely related formulations. “Geometric criticality” denotes a family of research programs in which critical behavior is generated by geometry, encoded in geometric observables, or constrained by geometric structures. In the most literal formulation, an underlying geometry changes discontinuously or loses a degree of freedom, and the critical response follows even when the corresponding scalar law varies continuously; the $p$-adic construction on $\mathbb{Q}_p^2$ provides a particularly explicit example, where collapse of a two-scale Pontryagin filtration at $h=1$ produces jumps in coordinate diffusion constants although the radial jump law depends continuously on $h$ (Rajkumar et al., 29 Sep 2025). In other literatures, the same term refers to singular Berry phases and quantum-geometric tensors, curvature singularities of information manifolds, configuration-space statistical geometry, structural transitions in scale-invariant networks, percolative geometry in active matter, curvature-driven entropy flow on constant-energy shells, and geometric responses of black-hole observables (Kartik et al., 6 May 2025, Kumar et al., 2014, Liu et al., 1 Aug 2025, Lucarini et al., 15 Jul 2025, Bhowmick et al., 5 Jun 2025, Cairano, 1 Dec 2025, Wu et al., 24 Jan 2026). The collected literature suggests that the expression is therefore best understood as a cross-disciplinary label for critical phenomena whose decisive structure is geometric rather than exclusively local or spectral.

1. Scope and principal meanings

Current usage falls into several distinct but overlapping categories. In some works, geometry is the mechanism of the transition; in others it is the diagnostic; in still others it is the constraint that forbids a trivially gapped or smoothly connected phase.

Context	Geometric object	Critical signature
Ultrametric random walks	Two-scale Pontryagin filtration on $\mathbb{Q}_p^2$	Jump in coordinate diffusion constants at filtration collapse (Rajkumar et al., 29 Sep 2025)
Quantum geometry	Berry phase, QGT, Bloch-angle manifold, geodesics	Divergent derivatives, gap closing, abrupt basis rotation (Kartik et al., 6 May 2025, Bao et al., 16 Dec 2025)
Information/configuration geometry	Parameter manifold, Fisher metric, Hamming-distance manifold	Curvature singularity, universal exponents, Fisher-information peak (Kumar et al., 2014, Liu et al., 1 Aug 2025)
Structural geometry	Network Laplacian RG, interaction graph, percolation clusters	Structural phase transition, fractal-dimension flow, evolving critical generations (Lucarini et al., 15 Jul 2025, Gunning et al., 18 Jun 2026, Wei et al., 23 Nov 2025)

This multiplicity is substantive rather than terminological. The ultrametric framework of geometry-induced criticality isolates a structural mechanism in which only the geometry changes (Rajkumar et al., 29 Sep 2025). By contrast, the non-Hermitian Kitaev-chain study uses “geometric criticality” for scaling of local quantum geometry near topological transitions (Kartik et al., 6 May 2025). Information geometry defines geometric critical exponents on parameter manifolds (Kumar et al., 2014), while configuration-space statistical geometry characterizes phase transitions through the distribution of pairwise distances between configurations (Liu et al., 1 Aug 2025). Network and sparse-graph studies use the term for topology-driven changes of effective dimension or connectivity (Lucarini et al., 15 Jul 2025, Gunning et al., 18 Jun 2026).

A plausible implication is that no single invariant definition spans all usages. What is common is the claim that the decisive nonanalyticity is visible in a geometric object: a filtration, curvature, metric tensor, manifold, graph, shell, or optimal measurement basis.

2. Geometry as a direct generator of critical behavior

The clearest direct mechanism is given by the $p$-adic random-walk construction on $\mathbb{Q}_p^2$ (Rajkumar et al., 29 Sep 2025). For $h\in(0,1]$, the weighted max-norm

$\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$

induces a duality-compatible two-scale filtration with radii $p^k$ and $\mathbb{Q}_p^2$0. The resulting shell-uniform random walk has two characteristic scales, and its scaling limit is a càdlàg anisotropic $\mathbb{Q}_p^2$1-adic Lévy process whose Fourier symbol is $\mathbb{Q}_p^2$2 and whose generator is the anisotropic Vladimirov operator $\mathbb{Q}_p^2$3 (Rajkumar et al., 29 Sep 2025). The critical point is $\mathbb{Q}_p^2$4: the two radii coincide, the fine filtration collapses into the coarse one, shell combinatorics must be re-merged and re-weighted uniformly, and the coordinate diffusion constants jump to the common isotropic value even though the radial convolution semigroup on shells remains continuous. The paper identifies this as the essence of “geometry-induced criticality” (Rajkumar et al., 29 Sep 2025).

The same paper states that the mechanism should extend to any second countable LCAG with a compact open subgroup. Introducing a two-scale perturbation of a regular Pontryagin filtration yields hierarchical random walks with alternating shell weights, and equalizing the two scales forces a re-indexing of shells together with a jump in component diffusion constants, while the radial law remains continuous (Rajkumar et al., 29 Sep 2025). This suggests a structural route to criticality in ultrametric models that does not require tuning a local analytic parameter.

A related “effective-geometry principle” is formulated by Gunning et al. for sparse long-range quantum lattice models on graphs of degree $\mathbb{Q}_p^2$5 (Gunning et al., 18 Jun 2026). There the ground-state phases and critical points are governed by the large-scale connectivity of the interaction graph. For even-$\mathbb{Q}_p^2$6 power-of-$\mathbb{Q}_p^2$7 graphs, the classical critical point is

$\mathbb{Q}_p^2$8

marking the change in dominant-shell geometry; for Fibonacci graphs, a single classical transition at $\mathbb{Q}_p^2$9 appears for odd $h=1$0, associated with a reordering that exchanges short- and long-bond geometry under $h=1$1 (Gunning et al., 18 Jun 2026). The criticality is therefore attached to graph geometry rather than to a local symmetry-breaking term.

Two microcanonical formulations push this idea further. In mean-field rotor Hamiltonians, the trace of the Weingarten operator on the constant-energy shell admits a universal collective expansion whose quadratic form has eigenvalues $h=1$2; geometric criticality is the vanishing of one of these curvature coefficients, $h=1$3, selecting the critical channel directly from shell geometry (Cairano, 30 Mar 2026). In “Phase Transitions as Emergent Geometric Phenomena,” entropy obeys the deterministic equation

$h=1$4

with $h=1$5 built from curvature invariants of the constant-energy manifold, so thermodynamic nonanalyticities arise from singularities or sign changes of a geometric source term (Cairano, 1 Dec 2025).

3. Quantum-state geometry, Berry phases, and geometric observables

A major line of work uses geometric phases and quantum geometry as critical probes. In the anisotropic XY chain under linear quench, the Berry phase per mode is

$h=1$6

and the total ground-state Berry phase develops a nonanalytic cusp at the critical field; in the XX limit $h=1$7, the mode phases become a sequence of non-contractible geometric phases, which the paper identifies as the geometric signature of the extended XX critical region (Sarkar et al., 2011). Closely related results were obtained for coupled optical cavity arrays modeled by an anisotropic Heisenberg spin-$h=1$8 lattice under linear quench, where the Berry-phase susceptibility develops a non-analytic cusp as $h=1$9, and summing over $h$0 yields the XX critical exponent $h$1 (Sarkar, 2013).

In the non-Hermitian long-range Kitaev chain, the derivative of the geometric phase and the quantum geometric tensor distinguish several universality classes (Kartik et al., 6 May 2025). For $h$2 and for $h$3 at $h$4, one obtains $h$5 and

$h$6

whereas for $h$7 at $h$8 the peak in $h$9 does not diverge and the Wannier-state correlation function develops a non-decaying plateau $\mathbb{Q}_p^2$0, which the paper interprets as an anomalous universality class (Kartik et al., 6 May 2025). Near exceptional points, the metric component $\mathbb{Q}_p^2$1 diverges with $\mathbb{Q}_p^2$2 on the Hermitian-Hermitian boundary and with $\mathbb{Q}_p^2$3 on the topological-coalescing boundary, defining a new EP universality class (Kartik et al., 6 May 2025).

The driven Jaynes–Cummings model provides an eigenstate-level version of the same theme (Chen et al., 8 Feb 2026). With drive amplitude $\mathbb{Q}_p^2$4 and phase $\mathbb{Q}_p^2$5 as control parameters, the quantum metric and Berry curvature of each eigenstate diverge near the photon-blockade-breakdown critical point. Because quasienergy gaps scale as $\mathbb{Q}_p^2$6, the leading singularity is

$\mathbb{Q}_p^2$7

and the divergence is stronger for bright states than for the unique dark state (Chen et al., 8 Feb 2026).

Quantum geometry also appears through geodesics and geometric defects. In the many-body analysis of precursors of criticality, finite-$\mathbb{Q}_p^2$8 diabolical points are isolated geometric singularities whose local geometry is that of a Bloch-sphere puncture with $\mathbb{Q}_p^2$9 and angular divergence; they act as “seeds of irregular geodesics,” and chains of such defects can condense along a first-order separatrix in the thermodynamic limit (Střeleček et al., 2024). In a different direction, the study of optimal Bell operators distinguishes “geometric criticality” from “geometric locking”: in the former, the optimal Bloch angles $p$0 jump discontinuously with simultaneous cusp in $p$1, divergence of $p$2, and nonlocal-gap closing; in the latter, a strong anisotropy pins one angle so that spectral indicators remain critical while the optimal basis geometry stays fixed (Bao et al., 16 Dec 2025).

4. Information geometry, configuration-space geometry, and bounded domains

Information geometry supplies one of the most systematic uses of the term. The parameter manifold with coordinates $p$3 carries metric

$p$4

and near a curvature singularity of a two-dimensional manifold the Ricci scalar and affine parameter scale as

$p$5

defining geometric critical exponents $p$6 and $p$7 (Kumar et al., 2014). The same exponents were found for the Van der Waals gas, Curie–Weiss ferromagnet, one-dimensional Ising chain, transverse XY chain, and RN–AdS black hole, which the paper presents as evidence for universality across classical and quantum models (Kumar et al., 2014).

Configuration-space statistical geometry shifts attention from parameter space to the ensemble of sampled configurations (Liu et al., 1 Aug 2025). For Ising variables, the normalized Hamming distance

$p$8

has variance determined by magnetization and two-point correlators. At a $p$9 symmetry-breaking critical point with $\mathbb{Q}_p^2$0 and $\mathbb{Q}_p^2$1, the principal scaling law is

$\mathbb{Q}_p^2$2

while in the orthogonal $\mathbb{Q}_p^2$3 basis, where no long-range order exists, the scaling reverts to the non-critical background

$\mathbb{Q}_p^2$4

(Liu et al., 1 Aug 2025). The same work defines a one-dimensional statistical manifold $\mathbb{Q}_p^2$5 with Fisher information

$\mathbb{Q}_p^2$6

and finds that $\mathbb{Q}_p^2$7 exhibits a sharp peak near $\mathbb{Q}_p^2$8 and is basis-independent within statistical error (Liu et al., 1 Aug 2025).

Gori and Trombettoni formulate bounded critical phenomena through a space-dependent scale factor $\mathbb{Q}_p^2$9 on a bounded domain $h\in(0,1]$0, with metric $h\in(0,1]$1 (Gori et al., 2019). In the interacting case, $h\in(0,1]$2 is determined by the Fractional Yamabe Equation, and the one-point correlator obeys

$h\in(0,1]$3

For the three-dimensional Ising model on a slab, fitting Monte Carlo data to this geometric form yields

$h\in(0,1]$4

in agreement, to five decimals, with the conformal-bootstrap estimate quoted in the paper (Gori et al., 2019). The same framework predicts that two-point functions depend on the fractional Q-hyperbolic distance computed from the uniformizing metric (Gori et al., 2019).

5. Structural transitions, percolation, and evolving geometric criticality

In scale-invariant networks, geometric criticality is formulated through the Laplacian density matrix $h\in(0,1]$5, the entropy $h\in(0,1]$6, and the entropic susceptibility

$h\in(0,1]$7

A plateau $h\in(0,1]$8 signals a well-defined spectral dimension, and infinitesimal structural perturbations can drive a topological phase transition at which the ultraviolet peak disappears and the notion of a single spectral dimension breaks down (Lucarini et al., 15 Jul 2025). Poggialini et al. report finite-size scaling

$h\in(0,1]$9

with, for the two-dimensional square lattice under rewiring, $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$0 and $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$1 (Lucarini et al., 15 Jul 2025). Beyond the threshold, the correlation dimension $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$2 decreases continuously, and in diluted Dorogovtsev–Goltsev–Mendes networks a hidden flow toward a Barabási–Albert fixed point appears around $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$3 (Lucarini et al., 15 Jul 2025).

Wei et al. describe a different geometric mechanism in iterative bicolored percolation (Wei et al., 23 Nov 2025). Starting from a critical two-state configuration, each generation independently recolors clusters and merges neighboring clusters of the same color, thereby deleting a subset of boundaries while preserving criticality. The result is a hierarchy of distinct but critical generations with exact generation-dependent fractal dimensions derived from CLE. For the symmetric case, the one-arm exponent is

$\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$4

and the cluster fractal dimension is

$\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$5

(Wei et al., 23 Nov 2025). Site and bond percolation follow different trajectories because of distinct initial conformal data and two-state structures (Wei et al., 23 Nov 2025).

Percolative geometry also organizes the phase transition of interacting run-and-tumble particles (Bhowmick et al., 5 Jun 2025). Dense clusters provide the geometric signature of motility-induced phase separation, with order parameter $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$6, susceptibility $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$7, and second-moment correlation length $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$8 (Bhowmick et al., 5 Jun 2025). Along the critical line, exponents vary continuously—for example, at $\|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p)$9, $p^k$0, $p^k$1, and $p^k$2, while near $p^k$3 they are $p^k$4, $p^k$5, and $p^k$6—yet the Binder-cumulant scaling function versus $p^k$7 coincides with the equilibrium two-dimensional lattice-gas Ising-percolation class (Bhowmick et al., 5 Jun 2025). The paper characterizes this as Ising-like super universality.

6. Continuum, anomaly-based, and spacetime formulations

Dascaliuc and Grujić introduced a geometric criticality scenario for the three-dimensional Navier–Stokes equations based on the linear sparseness of regions of intense vorticity (Dascaliuc et al., 2012). The criterion is scale-invariant because the relevant transverse scale is

$p^k$8

the reciprocal of the maximum vorticity (Dascaliuc et al., 2012). Combined with statistical positivity of ensemble-averaged vortex stretching across scales, this supports a filament scenario in which high-vorticity regions are long and thin, with one-dimensional sparseness at precisely the critical scale required to prevent blow-up (Dascaliuc et al., 2012). Here geometry is neither a diagnostic metric nor an order parameter; it is a measure-theoretic regularity mechanism.

Li and Yao give a geometric explanation of criticality in self-dual symmetry-protected topological models (Li et al., 2022). On a torus with symmetry-twisted boundary conditions, the twisted Hamiltonian remains invariant under a modified duality operator that anticommutes with another symmetry, forcing at least two-fold degeneracy of every eigenstate. Through spectral robustness, this implies that the self-dual theory cannot have a unique symmetric gapped ground state under periodic boundary conditions; equivalently, the combined symmetry and duality carry a mixed ’t Hooft anomaly (Li et al., 2022). In this usage, geometric criticality is linked to topology of boundary conditions and anomaly inflow rather than to curvature or metric singularities.

Wu, Shi, Li, and Yin extend the notion to observables of spacetime geometry (Wu et al., 24 Jan 2026). For static, spherically symmetric spacetimes, compact conserved perturbations of the stress-energy tensor induce a linear metric response

$p^k$9

with $\mathbb{Q}_p^2$00-bounded kernels, and the first-order shadow shift obeys

$\mathbb{Q}_p^2$01

Under mild assumptions on matter susceptibilities near a critical point, dominated convergence transfers the thermodynamic exponent to the geometric susceptibility,

$\mathbb{Q}_p^2$02

with controlled analytic corrections (Wu et al., 24 Jan 2026). The result turns black-hole shadow radius and photon-sphere frequency into geometric channels for critical response.

7. Unifying themes and recurrent distinctions

The literature suggests several recurrent distinctions.

First, geometric criticality does not always mean that a geometric observable diverges. In the $\mathbb{Q}_p^2$03-adic Lévy setting, the radial law remains continuous while coordinate diffusion constants jump because shell geometry collapses (Rajkumar et al., 29 Sep 2025). In Bell-operator optimization, spectral indicators may show conventional critical peaks while the basis remains geometrically locked (Bao et al., 16 Dec 2025). In configuration-space geometry, $\mathbb{Q}_p^2$04 is basis-sensitive, whereas Fisher information on $\mathbb{Q}_p^2$05 remains basis-independent within error (Liu et al., 1 Aug 2025).

Second, the term does not refer only to Berry phases. Berry-phase singularities and QGT divergences are central in quantum spin chains, cavity arrays, non-Hermitian topological systems, and the driven Jaynes–Cummings model (Sarkar et al., 2011, Sarkar, 2013, Kartik et al., 6 May 2025, Chen et al., 8 Feb 2026). But equally explicit uses concern ultrametric filtrations (Rajkumar et al., 29 Sep 2025), network fixed points and effective dimensions (Lucarini et al., 15 Jul 2025), energy-shell curvature instabilities (Cairano, 30 Mar 2026), geometric entropy flow (Cairano, 1 Dec 2025), and black-hole imaging observables (Wu et al., 24 Jan 2026).

Third, some frameworks treat geometry as cause, some as encoding, and some as constraint. The collapse of a two-scale filtration or the reorganization of an interaction graph is causal in the model definition (Rajkumar et al., 29 Sep 2025, Gunning et al., 18 Jun 2026). Information geometry and bounded-domain geometry encode criticality in curvature or scale factors (Kumar et al., 2014, Gori et al., 2019). Twisted-boundary anomaly arguments constrain the phase structure by excluding a trivially gapped state (Li et al., 2022).

A plausible overall conclusion is that geometric criticality has become an umbrella concept for situations in which the decisive singular structure is best formulated geometrically: as shell collapse, curvature singularity, graph reorganization, basis rotation, fractal-dimension flow, or anomaly-protected degeneracy. The recent ultrametric, network, active-matter, microcanonical, and spacetime formulations indicate that this viewpoint is no longer confined to quantum-state geometry, but is being used to connect criticality with topology, ultrametricity, sparse connectivity, and emergent geometry across a broad range of arXiv literatures (Rajkumar et al., 29 Sep 2025, Lucarini et al., 15 Jul 2025, Bhowmick et al., 5 Jun 2025, Cairano, 1 Dec 2025, Wu et al., 24 Jan 2026).