## 1 Introduction  
The rapid growth of Ethereum has exposed critical scalability challenges in its fundamental data structures, particularly the Merkle Patricia Trie (MPT) used for state management [1, 2]. Counting over 200 million unique addresses and growing, the efficiency of state storage and retrieval has turned into a bottleneck for network performance [3]. Most notably, according to Ethereum's founder, current MPT implementations can produce worst-case stateless proofs reaching up to 300 MB, significantly impacting client efficiency and network scalability [4].  
While MPTs have been extensively studied in theoretical computer science [5; 6], their specific application in blockchain systems presents unique challenges that remain inadequately addressed. Previous research has primarily focused on general trie properties or protocol-level optimizations [7; 8], leaving a significant gap in our understanding of MPT behavior at the blockchain scale.  
The fundamental challenge of state management in blockchain systems stems from the need to maintain a verifiable record of all account states while ensuring efficient access and updates. Merkle Patricia Tries (MPTs) were chosen as Ethereum's state management structure due to their unique combination of cryptographic verifiability and efficient key-value storage. However, as the network has grown to over 200 million accounts, the limitations of this approach have become increasingly apparent.  
A critical metric for MPT performance is path length--the number of nodes that must be traversed from root to leaf when accessing or proving a state. Path length directly impacts both storage requirements and proof sizes, with longer paths resulting in larger proofs and increased verification overhead. This relationship becomes particularly significant in the context of stateless clients, where every state access requires a complete Merkle proof.  
The current lack of precise path length characterization presents a significant obstacle to optimization efforts. While asymptotic bounds on trie properties are well understood, practical improvements require exact probability distributions and validated predictions across real-world scales. Our research addresses this gap through a rigorous mathematical analysis of MPT path lengths, providing both theoretical insights and practical optimization guidelines.  
This study presents a comprehensive probabilistic analysis of path length distributions in MPTs containing Ethereum addresses. Our primary contributions are:  
1. A strict mathematical model describing the distribution of path lengths in tries that contain random blockchain addresses;
2. Empirical validation by extensive computational experiments ranging from tries over 100 to 300 million addresses;
3. Accurate Prediction of Structural Properties: The experimental validation shows discrepancies that are not greater than 0.01 at all the tested scales;
4. Guidelines on practical implementation, showing potential proof size reductions of up to 70% using optimized path structuring.  
Our analysis reveals that average path lengths scale logarithmically with the number of addresses, following the distribution:  
\[P(PL=k)=\left(1-\left(\frac{1}{16}\right)^{k}\cdot\frac{15}{16}\right)^{N}- \left(1-\left(\frac{1}{16}\right)^{k-1}\cdot\frac{15}{16}\right)^{N},\]  
where \(PL\) is the path length, \(k\) is the number of nodes traversed, and \(N\) is the number of addresses in the trie.  
The remainder of this paper is organized as follows: Section 2 reviews relevant literature and theoretical foundations. Section 3 presents our probabilistic model. Section 4 describes the experimental methodology. Section 5 presents results and validation. Section 6 discusses implications and optimization strategies. Section 10 concludes with directions for future research.