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Anchored Nash inequalities and heat kernel bounds for a class of random conductance models with long-range jumps

This paper establishes anchored Nash inequalities for discrete non-local operators with degenerate weights to derive on-diagonal heat kernel upper bounds for random conductance models featuring long-range jumps on integer lattices and supercritical percolation clusters.

Original authors: Sebastian Andres, Xin Chen, Martin Slowik, Kun Yin

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sebastian Andres, Xin Chen, Martin Slowik, Kun Yin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a drop of ink spread through a glass of water. Sometimes the water is clear and the ink spreads smoothly and predictably, like a perfect circle. Other times, the water is full of obstacles—maybe thick jelly, or a maze of tangled spaghetti, or even holes where the ink can't go at all. In those messy situations, the ink doesn't spread in a simple circle; it gets stuck, it jumps over gaps, and it moves in weird, unpredictable ways. Scientists who study how things move through messy environments call this "random walks." They want to know: if I drop a particle here, how likely is it to be found there after a certain amount of time? This question is crucial for understanding everything from how heat moves through a broken-down engine to how electricity flows through a damaged wire, or even how a virus might spread through a crowded city.

To answer these questions, mathematicians use a special tool called a "Nash inequality." Think of this as a rulebook that predicts how fast the ink (or heat, or a particle) can spread. If the environment is uniform and smooth, the rulebook is simple. But if the environment is "degenerate"—meaning some paths are super wide and others are super narrow, or some are blocked entirely—the old rulebook breaks down. For a long time, scientists had a way to fix this for simple, short-step walks (where a particle only moves to its immediate neighbor), but they struggled when particles could make giant "long-range jumps," leaping over huge gaps in the maze. This paper tackles that exact problem: how to write a new, robust rulebook for particles that can both shuffle slowly and leap far, even when the ground they walk on is a chaotic, broken mess.

The authors of this paper, Sebastian Andres, Xin Chen, Martin Slowik, and Kun Yin, have developed a new version of this rulebook called an "anchored Nash inequality." Imagine you are trying to describe how a crowd of people moves through a stadium. In a normal stadium, everyone moves at a similar speed. But in a broken stadium, some aisles are wide, some are narrow, and some are blocked. If you try to predict the movement from the center of the stadium, you might get it wrong because the center is special. The "anchored" part of their new rule means they fix their perspective to a specific starting point (like the main entrance) and measure everything relative to that spot. This allows them to handle the chaos of the broken stadium without getting confused.

They proved that even if the "weights" of the paths (how easy or hard it is to walk on them) are wildly different—some paths are incredibly easy, others are nearly impossible, and some are so hard they might as well be infinite—this new anchored rule still works. They showed that as long as the "messiness" of the environment isn't too extreme (specifically, if the average difficulty doesn't blow up in a specific mathematical way), you can still predict how the particle spreads.

The paper specifically looks at two types of movement:

Short steps: Moving only to the very next neighbor.
Long-range jumps: Leaping over many neighbors at once, like a frog jumping over a pond.

They found that for these long-range jumps, the "anchored" rule is essential. Without it, the math falls apart when the environment is too irregular. By using this new inequality, they were able to calculate "heat kernel bounds." In plain English, this is a way to say, "Here is the maximum speed at which the ink can spread." They proved that even in these chaotic, long-jumping scenarios, the spread of the particle follows a predictable pattern (specifically, it decays like $t^{-d/2}$ , where $t$ is time and $d$ is the number of dimensions) as long as the environment meets certain conditions.

The authors didn't just stop at theory; they applied their findings to two very real-world scenarios. First, they looked at "ergodic environments," which are like random mazes where the rules of the maze are the same everywhere on average, but locally they look different. Second, they applied it to "supercritical percolation clusters." Imagine a giant sponge where some holes are filled with water and others are air. If you have enough water, the wet parts connect to form one giant, giant island. This paper shows that even on this giant, irregular island, if a particle can jump long distances, we can still predict how it moves.

One of the most exciting parts of their discovery is that they didn't need the paths to be "uniformly elliptic." In the old days, mathematicians required that every path be at least somewhat easy to walk on. This paper says, "Nope, we don't need that." Even if some paths are effectively impossible (infinite resistance), as long as the average behavior of the environment is under control, the prediction holds. They also showed that their method works for "annealed" bounds, which means they can predict the average behavior of the particle across all possible versions of the messy maze, not just one specific maze.

The paper is careful to note what it doesn't do. It doesn't give a perfect, two-sided prediction for every single point in the maze (near-diagonal bounds) in all dimensions, especially in two dimensions where things get tricky. It also doesn't claim that the particle moves exactly like a standard bell curve (Gaussian) in every case; for long-range jumps, the behavior is different. However, for the specific question of "how fast can it spread from the starting point?", they have provided a solid, proven upper limit.

In summary, this paper builds a new mathematical safety net. It catches the behavior of particles that jump long distances through broken, irregular worlds. By anchoring their calculations to a starting point and using a clever new inequality, the authors proved that even in the most chaotic, degenerate environments, there is still a limit to how fast things can spread. This gives scientists a reliable tool to model complex systems, from electricity in damaged materials to the spread of information in social networks, ensuring that even when the world is messy, the math can still make sense.

Technical Summary: Anchored Nash Inequalities and Heat Kernel Bounds for a Class of Random Conductance Models with Long-Range Jumps

Problem Statement
The paper addresses the challenge of establishing heat kernel bounds for random walks on graphs where the jump rates (conductances) are degenerate, meaning they can be arbitrarily close to zero or unbounded from above. Specifically, the authors extend the analysis beyond nearest-neighbor random walks to include models with long-range jumps.

In the context of Random Conductance Models (RCM), standard Nash inequalities—which relate the $L^2$ -norm of a function to its Dirichlet form and $L^1$ -norm—often fail when weights are not uniformly elliptic (i.e., not bounded away from zero). While "anchored" Nash inequalities were previously introduced by Mourrat and Otto for nearest-neighbor walks with degenerate weights, this work tackles the more complex scenario where:

The underlying graph may be a general locally finite graph, including supercritical percolation clusters.
The random walk admits long-range jumps (non-local operators).
The reference measure (speed measure) is general and not necessarily the counting measure.
The weights satisfy only specific integrability conditions rather than uniform bounds.

Methodology
The authors employ a functional analytic approach centered on the derivation and application of anchored Nash inequalities.

Anchored Nash Inequality: The core technical achievement is proving an anchored Nash inequality for non-local operators. Unlike the standard Nash inequality, the anchored version controls the $L^2$ -norm of a function $f$ using a weighted Dirichlet form, the $L^1$ -norm, and a weighted $L^2$ -norm involving a term $\|\eta^{m/2}f\|_2$ , where $\eta(y) = \max\{d(o, y), 1\}$ . This breaks translation invariance but allows for control in degenerate settings.
Interpolation: The authors derive an "interpolated" version of this inequality, which introduces exponents $\alpha, \beta, \gamma$ summing to 1, allowing for a more flexible balance between the Dirichlet form, the $L^1$ -norm, and the weighted $L^2$ -norm.
Time-Change Technique: To handle the long-range jumps and general reference measures, the authors introduce a time-change of the original process $X_t$ . They define a new process $Y_t$ with a speed measure $\pi_{m_0} = \max\{1, \mu_{m_0}, \theta\}$ , where $\mu_{m_0}$ captures the moments of the jump lengths. This time-changed process satisfies conditions (specifically regarding the decay of long-range jumps) that allow the application of the Nash inequality to derive heat kernel bounds.
Transfer of Bounds: Using the relationship between the original process and the time-changed process (via an additive functional), the authors transfer the on-diagonal upper bounds derived for $Y_t$ back to the original process $X_t$ .
Percolation Verification: For supercritical percolation clusters, the authors verify that the necessary geometric conditions (volume doubling and Sobolev-type inequalities) hold on large scales, utilizing known results on the comparability of graph distance and Euclidean distance on these clusters.

Key Contributions and Results

Theorem 1.3 & 1.4 (Anchored Nash Inequalities): The paper establishes an anchored Nash inequality and its interpolated version for a general class of graphs satisfying Assumption 1.1 (volume growth and Sobolev-type regularity). These inequalities hold for general reference measures $\theta$ and weights that are unbounded from above and below, provided specific integrability conditions on $\theta$ , $\theta^{-1}$ , and the jump moment measures are met.
Theorem 1.5 (On-Diagonal Heat Kernel Bounds): Using the derived inequalities, the authors prove quenched on-diagonal upper bounds of the form $p(t, o, o) \leq C t^{-d/2}$ $p (t, o, o) \leq C t^{- d /2}$ .
- This result applies to random walks with long-range jumps, provided the jump kernel is $L^2$ -integrable and controlled by the integrability of the measure $\mu_{m_0}$ .
- The bounds hold for dimensions $d \geq 3$ .
- The constants in the bounds depend on the specific realization of the environment (quenched) but are explicit in terms of the moments of the conductances.
Theorem 1.8 & 1.9 (Ergodic Environments): The results are applied to random conductance models in ergodic environments.
- Quenched: The heat kernel bound holds almost surely for a fixed environment.
- Annealed: By taking expectations, the authors derive annealed heat kernel bounds. For $d \geq 3$ , the annealed bound scales as $t^{-d/2}$ . For $d=2$ , a logarithmic correction appears, yielding a bound of order $(1 + \log t)t^{-1}$ .
Theorem 1.14 & 1.15 (Percolation Clusters): The framework is extended to supercritical percolation clusters (including those with long-range correlations). The authors establish the first global version of a functional inequality (anchored Nash) on these degenerate models and derive corresponding heat kernel bounds.

Significance and Claims
The paper claims to extend the scope of functional inequality methods in random environments in several ways:

Long-Range Jumps: It is one of the first works to successfully apply anchored Nash inequalities to random walks with long-range jumps that still exhibit Gaussian large-scale behavior. Previous methods (e.g., Davies' perturbation or Moser iteration) were noted as difficult to apply directly to long-range jumps due to the variety of jump lengths.
General Reference Measures: The results remove the restriction of using the counting measure, accommodating general speed measures $\theta$ , which is crucial for models like the Variable Speed Random Walk (VSRW) and Constant Speed Random Walk (CSRW).
Degenerate Weights: The approach handles weights that are not uniformly elliptic, relying instead on moment conditions. This is significant for models like percolation clusters where conductances can vanish.
Explicit Constants: The authors highlight that their method yields explicit random constants in the heat kernel bounds. This explicitness facilitates the derivation of annealed estimates (averaged over the environment) under relatively mild moment conditions, contrasting with other techniques where annealed results require more complex or intrinsic moment assumptions.
Limitations: The authors modestly note that their bounds are strictly on-diagonal ( $p(t, o, o)$ ). They state that deriving effective near-diagonal bounds ( $p(t, o, x)$ ) is currently hindered because the constants depend on the fixed origin $o$ . Consequently, the results are established for dimensions $d \geq 3$ ; the case $d=2$ requires a different treatment (logarithmic correction) and is not covered by the standard Gaussian scaling in the same way.

The work provides a robust analytical toolkit for studying the heat kernel of degenerate, non-local random walks, bridging the gap between nearest-neighbor models and $\alpha$ -stable regimes while maintaining Gaussian scaling under suitable moment conditions.

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