Upper-Tail Accumulation Scale

• Upper-tail accumulation scale is a measure that defines the level at which rare events in random systems become non-perturbative and display large deviation phenomena.
• It quantifies the excess above the mean through forms such as fluctuation magnitudes, rate function exponents, or localization windows, with examples like N^(1/3) in KPZ models and n^2 in first passage percolation.
• Understanding this scale is essential for developing precise probabilistic models and statistical tests, as it reveals the underlying geometry and dependence structures of various complex systems.

Searching arXiv for recent and foundational papers related to “upper-tail accumulation scale” across the domains represented in the source material. Upper-tail accumulation scale denotes the characteristic scale on which a random system realizes unusually large values. In the literature represented here, the notion appears in three closely related forms: the size of the excess above the mean that supports nontrivial upper-tail mass, the speed governing the logarithm of the tail probability, and the geometric or pathwise window in which a conditioned rare event localizes. In stationary KPZ models, upper-tail events occur on the scale $tN^{1/3}$ and decay as $e^{-c t^{3/2}}$; in first passage percolation they occur at the area scale $n^2$; in sparse random graphs they are governed by entropic scales such as $n^2 p^\Delta \log(1/p)$ (Landon et al., 2022, Basu et al., 2017, Bhattacharya et al., 2019).

1. Core meanings of the scale

The term is not tied to a single normalization across disciplines. Rather, the common content is that it identifies the level at which the upper tail ceases to be perturbative and acquires its characteristic large-deviation form. In some models this is a fluctuation magnitude, in others a rate function scale, and in others a localization scale under conditioning.

Setting	Characteristic upper-tail scale	Representative form
Stationary KPZ models	$sN^{1/3}$	$$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$
First passage percolation	$n^2$	$$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$
Sparse subgraph upper tails	$n^2 p^\Delta \log(1/p)$	variational upper-tail rate
Upper-tail KPZ path measures	$L^{-1/4}$ and $e^{-c t^{3/2}}$0	Brownian bridge limit and quenched localization

This suggests that “accumulation” should be read structurally rather than literally. It may refer to where probability mass accumulates above the mean, to the exponent in the log-probability, or to the window in which a conditioned extremal configuration concentrates (Landon et al., 2022, Basu et al., 2017, Bhattacharya et al., 2019, Ganguly et al., 2023).

2. KPZ-class models and conditioned geometry

For stationary integrable directed polymer models with boundaries and for a family of interacting Brownian diffusion models at equilibrium, Landon and Sosoe prove upper tail bounds of the expected KPZ form. The essential statement is

$e^{-c t^{3/2}}$1

with the same structure for the Brownian height function, for $e^{-c t^{3/2}}$2 up to order $e^{-c t^{3/2}}$3. Their summary explicitly characterizes the upper-tail accumulation scale as $e^{-c t^{3/2}}$4: the mass of the upper tail probability accumulates above the mean at deviations $e^{-c t^{3/2}}$5, while the exponent in the probability is proportional to $e^{-c t^{3/2}}$6. The proof proceeds through the Rains-EJS exponential identity, together with monotonicity and convexity, and the same summary notes that the corresponding lower-tail bounds obtained by this method are sub-optimal in the exponent (Landon et al., 2022).

At the continuum KPZ level, the directed landscape exhibits a metric large deviation principle at speed $e^{-c t^{3/2}}$7. Metrics of finite rate are in one-to-one correspondence with planted network measures supported on a finite or countable collection of paths, and the rate function is

$e^{-c t^{3/2}}$8

For the directed geodesic, the resulting upper-tail rate is cubic: $e^{-c t^{3/2}}$9 Here the accumulation scale is no longer a fluctuation magnitude around the mean of a scalar observable, but the speed of a metric-level LDP and the cubic scale of path deviation (Das et al., 2024).

Conditioning the KPZ fixed point with narrow-wedge initial data on a large value at a specific point produces a further refinement. Under the conditioning $n^2$0, the local field is rescaled by

$n^2$1

and the finite-dimensional distributions converge to an upper tail field defined on the full two-dimensional space. In the negative time regime this field behaves like a Brownian-type field, while in the positive time regime it behaves like the KPZ fixed point. One main ingredient is a joint upper-tail estimate generalizing the one-point GUE Tracy-Widom upper tail (Liu et al., 1 Jan 2025).

For path measures, conditioning on large passage time or free energy changes the transversal scaling. In both zero and positive temperature KPZ models, the rescaled geodesic or polymer converges to a standard Brownian bridge after scaling by $n^2$2, so the path fluctuates on the smaller spatial scale $n^2$3. In the positive temperature case, the polymer further localizes around a random backbone within a window of width $n^2$4 up to logarithmic corrections. This suggests a hierarchy of upper-tail scales in conditioned KPZ geometry: one-point excess, local profile window, annealed path scale, and quenched localization scale (Ganguly et al., 2023).

3. Large-deviation speeds in random media and function spaces

In first passage percolation on $n^2$5 with i.i.d. bounded edge weights, the upper-tail event

$n^2$6

satisfies

$n^2$7

and Basu, Ganguly, and Sly prove that the normalized log-probability converges: $n^2$8 The heuristic given in the summary is that all parallel disjoint paths must be slow, so the event accumulates over area rather than along a single path. The same summary states that the method extends to higher dimensions, where the scale becomes $n^2$9, and to last passage percolation (Basu et al., 2017).

For $n^2 p^\Delta \log(1/p)$0-times integrated Brownian motion, the upper-tail accumulation mechanism is functional rather than combinatorial. Under the uniform norm, the upper tail is localized at the point of maximal variance, $n^2 p^\Delta \log(1/p)$1, and

$n^2 p^\Delta \log(1/p)$2

Under the $n^2 p^\Delta \log(1/p)$3 norm, the tail is governed by the largest eigenvalue $n^2 p^\Delta \log(1/p)$4 of the covariance operator: $n^2 p^\Delta \log(1/p)$5 The summary explicitly interprets the accumulation scale here as concentration on the extremal point for the supremum norm and on the leading eigenspace for the $n^2 p^\Delta \log(1/p)$6 norm (Gao et al., 2014).

For exponential functionals of subordinators,

$n^2 p^\Delta \log(1/p)$7

precise upper-tail equivalents are expressed in terms of the Laplace exponent $n^2 p^\Delta \log(1/p)$8 and the inverse function $n^2 p^\Delta \log(1/p)$9 of $sN^{1/3}$0. Under

$sN^{1/3}$1

the paper gives exact asymptotics for both $sN^{1/3}$2 and the density $sN^{1/3}$3, improving prior logarithmic asymptotics. In this setting, the relevant scale is encoded analytically by $sN^{1/3}$4, $sN^{1/3}$5, and the behavior of the Lévy measure near $sN^{1/3}$6, rather than by a single power of the system size (Haas, 2021).

4. Sparse graphs, additive structures, and the missing logarithm

For sparse homomorphism counts, the upper-tail accumulation scale is entropic. In sparse Erdős–Rényi graphs, and in the corresponding uniform and regular graph models, the relevant scale is

$sN^{1/3}$7

and the upper tail is described by a variational problem minimizing relative entropy subject to a homomorphism constraint. The summaries state that the rare event can be realized by planting a clique or planting a hub, while for joint upper tails of several graphs the optimum may involve both mechanisms simultaneously (Bhattacharya et al., 2019).

A central technical issue in this area is the “missing log” problem. Warnke’s BK-inequality-based sparsification method recovers the factor $sN^{1/3}$8 in settings where earlier Kim–Vu and Janson–Ruciński methods were sharp only up to that logarithm. The general form stated in the summary is

$sN^{1/3}$9

For arithmetic progressions, Schur triples, additive quadruples, and $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$0-sums, this determines the exponent up to constants and identifies the correct upper-tail accumulation scale in the sparse regime (Warnke, 2016).

For cycle counts in $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$1, the same logarithmic issue is resolved up to constants in the exponent. The main theorem summarized for $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$2 is

$$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$3

The summary explicitly says that the JOR lower bound is the truth for $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$4-cycles when $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$5, so here the accumulation scale switches between the entropic scale $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$6 and the global-count scale $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$7 (Raz, 2019).

For irregular graphs, Basak and Karmakar identify several regimes. In the localized regime $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$8, the normalized logarithmic upper tail is governed by $$\mathbb{P}(\text{centred variable} > sN^{1/3}) \leq C e^{-c s^{3/2}}$$9, and the typical conditioned graph contains a planted local structure. In the Poisson regime

$n^2$0

the upper tail is that of a sequence of Poisson random variables with diverging mean. For the $n^2$1-armed star $n^2$2, the intermediate regime has scale $n^2$3. These results make the accumulation scale explicitly regime-dependent, with transitions between localized planting and distributed Poissonian surplus (Basak et al., 7 Mar 2025).

Combinatorial proof methods also isolate the scale through high moments. Hązła and Holenstein introduce growth boundedness and prove

$n^2$4

for $n^2$5-growth bounded distributions, and they recover the Janson–Oleszkiewicz–Ruciński upper-tail bound for the number of copies of a fixed graph in an Erdős–Rényi random graph. In this line of work, the accumulation scale is the moment order $n^2$6 that still controls the relevant large deviation (Hązła et al., 2014).

5. Algorithmic, network, and aggregate-tail manifestations

In randomized algorithms, the upper-tail accumulation scale can be read directly from the exponent of the running-time tail. For Bucket Sort with a quadratic per-bucket routine, the paper proves

$n^2$7

whereas for the $n^2$8 variant and for trie external path length,

$n^2$9

The summaries interpret the difference by the way overload accumulates: moderate overload in roughly $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$0 buckets for the quadratic variant, versus additive accumulation across buckets or leaves for the $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$1 and trie cases (Bercea et al., 2020).

In scale-free inhomogeneous random graphs, the relevant upper-tail scale for inference is the number of upper order statistics that remain aligned. The paper proves that if

$$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$2

then the nodes with the largest $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$3 weights also have the largest $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$4 degrees, with matching order and high probability. This alignment transfers classical extreme value asymptotics to the degree sequence and yields asymptotic normality for the Hill, Pickands, and PWM estimators based on the upper degrees (Cirkovic et al., 2024).

For weighted sums of bivariate random variables, the upper-tail behavior of the aggregate depends on the marginal shape parameters and on extremal dependence. The summary states that if both marginal shape parameters are negative, the aggregate upper tail is determined by both marginal shape parameters and the coefficient of asymptotic independence; if they are both positive or have different signs, the upper-tail behavior of the aggregate is given solely by the largest marginal shape. In this setting, the accumulation scale is the effective tail parameter of the sum, and it may either preserve or thin the marginal upper tail depending on the dependence class (Richards et al., 2021).

6. Statistical scaling, testing, and variance-scale phenomena

In conditional EVT for returns, the scale is operationalized by horizon rescaling. After fitting an AR(1)-GARCH(1,1) model with Student-$$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$5 innovations and applying the Hill estimator to filtered residuals, the paper uses the EVT $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$6-root-of-time law: $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$7 The methodology targets the upper tail and explicitly contrasts this with the Gaussian square-root-of-time law. Here the upper-tail accumulation scale is the multi-period factor $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$8 applied to single-period tail quantities (Cotter, 2011).

A tail-sensitive goodness-of-fit statistic makes the scale tunable through a parameter: $$\mathbb{P}(T_n \ge (\mu+\zeta)n)=\exp(-\Theta(n^2))$$9 The summary states that larger $n^2 p^\Delta \log(1/p)$0 increases weight to the upper tail, so the degree of upper-tail accumulation emphasized by the test is explicitly controlled by $n^2 p^\Delta \log(1/p)$1. This differs from Kolmogorov-Smirnov-type procedures, which distribute sensitivity more evenly across the support (Meissner, 2012).

For Weibull-type light tails, the relevant scale is the upper-order sample size. The estimators of the Weibull tail-coefficient use only the $n^2 p^\Delta \log(1/p)$2 largest order statistics, with $n^2 p^\Delta \log(1/p)$3 and $n^2 p^\Delta \log(1/p)$4, through linear combinations of log-spacings. Their asymptotic normality shows that the informative part of the sample is concentrated in the extreme upper order statistics rather than across the full empirical distribution (Gardes et al., 2011).

A distinct number-theoretic instance appears in quadratic twists of elliptic curves. For central values $n^2 p^\Delta \log(1/p)$5, the upper-tail density is controlled at the variance scale

$n^2 p^\Delta \log(1/p)$6

and the paper proves a Gaussian upper bound matching random-matrix predictions at that scale. In this setting the accumulation scale is neither a power of a system size nor an order-statistic threshold, but the variance of the logarithm of the central value (Creighton, 11 Jul 2025).

Taken together, these works show that upper-tail accumulation scale is a comparative concept: it specifies where a rare-event mechanism becomes structurally visible. Depending on the model, that mechanism may be KPZ $n^2 p^\Delta \log(1/p)$7 fluctuation, area-order slowdown, entropy-order planting, Brownian-bridge localization, Poissonian surplus, or variance-scale Gaussian decay. The unifying question is the same in each case: on what scale does the upper tail stop looking incidental and start revealing the geometry, dependence structure, or optimization principle of the underlying model.