# Epistral Network: Revolutionizing Media Curation and Consumption through
Decentralization

Dipankar Sarkar
Epistral Protocol
<EMAIL_ADDRESS>
Shubham Upadhyay
Epistral Protocol
<EMAIL_ADDRESS>

###### Abstract

Blockchain technology has revolutionized media consumption and distribution in
the digital age, allowing creators, consumers, and regulators to participate
in a decentralized, fair, and engaging media environment. Epistral, an
innovative media network that leverages blockchain technology, aims to be the
world’s first anti-mimetic media curation and consumption network, addressing
the core challenges facing today’s digital media landscape: unfair treatment
of creators and manipulative consumer algorithms, and the complex task of
effective regulation. This paper delves into the conceptualization, design,
and potential impact of epistral and explores how it embodies McLuhan’s and
Girard’s theories within the realm of blockchain technology and draws from
Hayden’s critique of democratic representation. The paper analyzes the
challenges and opportunities presented by this new network, providing a
broader discourse on the future of media consumption, distribution, and
regulation.

_Keywords_ Blockchain $\cdot$ Social Media $\cdot$ Recommendation Systems
$\cdot$ Mimetic Theory

##  1 Introduction

In an era where digital media consumption is ubiquitous and constantly
evolving, the profound observation of Marshall McLuhan, "the medium is the
message," has never been more relevant. McLuhan’s assertion, originating from
his 1964 work "Understanding Media: The Extensions of Man," posits that the
medium through which content is conveyed fundamentally shapes and controls the
scale and form of human association and action (McLuhan, 1964). This
principle, while initially applied to the media landscapes of the mid-20th
century, finds renewed significance in the context of today’s digital
networks, particularly when examining the intersection of media, technology,
and society.

Drawing from McLuhan’s foundational concepts, Girard’s mimetic theory provides
an alternative perspective on contemporary media dynamics (Girard, 1965).
Girard posits that human desires are inherently imitative, originating from
others, thus creating a cycle of imitation and competition that is prevalent
in today’s culture driven by social media. In the context of modern internet
recommendation systems, there is a tendency for bias towards creating echo
chambers for users. These systems are designed to present users with content
similar to what they have previously encountered or engaged with, rather than
offering diverse or challenging content. Consequently, users may be exposed to
a limited range of ideas and opinions, reinforcing their existing beliefs and
attitudes. This phenomenon can have detrimental effects on the cultivation of
a well-informed and healthy society, as it can facilitate the dissemination of
misinformation and the perpetuation of established power structures.

In the contemporary digital age, the intersection of these theories becomes
increasingly relevant, especially when considering the challenges faced by
content creators, consumers, and regulators within the digital media
landscape. Content creators often grapple with unfair systems that exploit
their work without equitable compensation, while consumers are subjected to
aggressive algorithms designed to manipulate their attention and preferences.
Regulators, on the other hand, struggle to balance the need for free
expression with the necessity of maintaining societal norms and values in an
ever-changing digital environment.

Grant M. Hayden, in "The False Promise of One Person, One Vote," examines the
complexities of political equality in democratic systems, highlighting how
individual influence is often misrepresented or diluted in larger democratic
processes (Hayden, 2003). This discussion parallels the challenges in the
digital media landscape, where individual voices and content creators are
often overshadowed by larger entities or algorithms. Just as Hayden critiques
the democratic ideal of equal representation, the landscape of digital media
faces similar issues of unequal representation and influence.

The proposed solution to these challenges is Epistral, an innovative media
network that leverages blockchain technology. Blockchain, first conceptualized
by Satoshi Nakamoto in the creation of Bitcoin (Nakamoto, 2008), offers a
decentralized and transparent platform that inherently challenges the
traditional centralized models of content distribution and consumption. The
Steem network, a blockchain-based social media platform, serves as a technical
inspiration for Epistral due to its novel approach to content creation and
distribution (Steemit Inc., 2016).

Epistral aims to be the world’s first anti-mimetic media network, utilizing
blockchain not just as a technology but as a medium to foster a new paradigm
in content consumption and distribution. This approach seeks to address the
core challenges facing today’s digital media landscape: unfair treatment of
creators, manipulative consumer algorithms, and the complex task of effective
regulation. Unlike traditional platforms, Epistral’s network does not focus on
preventing intellectual property issues or serving as a store of value.
Instead, it focuses on providing decentralized tools that empower content
producers and facilitate a fair, engaging, and regulator-friendly media
environment.

In this light, Hayden’s insights provide a critical lens through which to
assess the potential of blockchain in addressing the democratic deficits in
digital media. We go beyond tackling the issues of how creators are treated
and how algorithms manipulate content. Our vision is a future where successful
regulation supports a flourishing media environment, while also maintaining
the core principles of opposing censorship, promoting free speech, and
embracing decentralization.

This paper delves into the conceptualization, design, and potential impact of
Epistral, exploring how it embodies McLuhan’s and Girard’s theories within the
realm of blockchain technology and draws from Hayden’s critique of democratic
representation. By analyzing the challenges and opportunities presented by
this new network, we aim to contribute to the broader discourse on the future
of media consumption and distribution in the digital age, as well as its
implications for political equality and democratic processes.

##  2 Background

The advent of blockchain technology has catalyzed a wave of innovation in
various sectors, notably in the realm of digital media. This section delves
into the background and inspiration behind the creation of Epistral,
highlighting key developments in blockchain-based media platforms that have
paved the way for this novel network.

Figure 1: Challenges in recommendation systems

#### Monetized social feeds & Polarisation of Society

The popularity and widespread use of social media platforms such as Facebook,
Instagram, and TikTok have grown significantly. These platforms offer users
the ability to create and share content with a large audience. However, they
also have the potential to create echo chambers, where users are only exposed
to content that aligns with their initial preferences. This is particularly
evident for users who are exposed to biased videos, as they are more likely to
encounter similar content in the future. As a result, there has been a
noticeable increase in the dissemination of misinformation videos, as users
are more likely to be exposed to content that reinforces their existing
beliefs. Furthermore, this limited exposure to diverse perspectives can lead
to a lack of understanding and empathy towards different viewpoints,
ultimately contributing to a more polarized society.

#### Emergence of Blockchain in Social Media

The integration of blockchain technology into social networks began as an
effort to address the centralization and control prevalent on traditional
platforms. The inherent characteristics of the blockchain of decentralization,
transparency, and immutability offered a new paradigm for content creation,
distribution, and monetization, fundamentally challenging existing media
models.

A variety of blockchain-powered media outlets have been created, each of them
providing distinctive perspectives and approaches that have been taken into
account in the formation of Epistral: a Decentralized Platform for Content
Aggregation and Incentivization.

D.tube, one of the pioneers in blockchain-based video platforms, emerged as a
decentralized counterpart to YouTube. Initially, it was built on the Steem
blockchain and utilized the InterPlanetary File System (IPFS) for
decentralized storage. Users were rewarded with STEEM tokens, which
demonstrates how the blockchain could incentivize content creation and
engagement. Subsequently, D.tube moved to its own blockchain Avalon, while
still on the HIVE network, and introduced the DTUBE coin, marking a
significant evolution in its approach.

In the realm of livestreaming, Dlive began its journey on the Steem
blockchain, but underwent several transitions. It moved to the Lino blockchain
and later to TRON, illustrating the fluid nature of blockchain applications in
media. Dlive’s journey is indicative of the broader search for optimal
operational frameworks within blockchain-based media platforms.

TravelFeed, originally a Steem ecosystem project, provided a curated blogging
platform focused on travel content. It later transitioned to the Hive
blockchain, allowing creators to use their own domains. This shift highlighted
the ongoing evolution of blockchain platforms, particularly in enhancing
content creators’ autonomy and monetization opportunities.

3Speak, a platform dedicated to live streaming and video content, emphasized
free speech. It was initially built on Steem (Steemit Inc., 2016) and IPFS
(Benet, 2014). However, following Steem’s acquisition by TRON, 3Speak, along
with many other Steem-based decentralized applications (dApps), migrated to
the Hive blockchain. This move highlighted the significance of community
governance and blockchain’s role in maintaining platform independence.

Finally, Dlike emerged as a decentralized blend of Medium and Instagram,
offering a platform for sharing social media posts. It featured elements like
trending content and topic filters, and its layout bore a resemblance to
Steemit’s. The existence of Dlike underlined the potential of blockchain
technology to replicate and enhance existing social media functionalities.

These platforms collectively demonstrate the versatility and transformative
potential of blockchain technology in various domains of online media, from
video streaming to social blogging, each adapting and evolving in its unique
way.

#### Epistral’s Conceptual Foundation

The experiences of these pioneering platforms have been instrumental in
shaping the vision of Epistral. The network draws on the successes and
challenges faced by its predecessors, with the aim of refine and advance the
application of blockchain in media. Epistral’s unique approach is further
inspired by the theories of McLuhan and Girard, integrating their insights
into the design and functionality of the network. By combining the technical
advancements demonstrated by platforms like D.tube, Dlive, TravelFeed, 3Speak,
and Dlike with a deep understanding of media theory, Epistral positions itself
as a groundbreaking entity in the blockchain media landscape.

In conclusion, Epistral is a product of the technological advances of
blockchain in media and the theoretical frameworks of McLuhan and Girard. This
combination is intended to tackle the current issues in media consumption,
providing a decentralized, equitable, and stimulating platform for creators,
consumers, and regulators.

##  3 The Epistral Network: Vision and Goals

The Epistral Network is a revolutionary response to the multifarious issues
and trends of the current digital media environment. Its purpose is to rethink
and redefine the way content is produced, distributed, and managed, taking
into account both the theoretical foundations of media experts and the
technical progress in blockchain technology.

At its core, Epistral Network is an algorithm company that focuses on
promoting credible, diverse, and enjoyable platform recommendations. The
credibility of these recommendations is determined by a combination of
reputation and personal investment, with each user interaction carrying a
unique value. Although the network is open to everyone, it effectively
addresses the issues of content discovery and incentivization by providing a
solution that caters to credible and enjoyable recommendations.

The vision of Epistral is to create a media network that surpasses traditional
boundaries and restrictions, creating an ecosystem where fairness, diversity,
and autonomy are not just ideals, but actual realities. The fundamental
objectives of the Epistral Network are multifaceted, each tackling a key
element of the media system:

  1. 1.

Diversity in Media Consumption: Epistral aims to break the echo chambers and
algorithmic biases prevalent in existing media platforms. The network is
designed to promote a diverse range of content, encouraging exposure to varied
perspectives and ideas. This goal is not only about offering a wide array of
content but also about ensuring that this content is accessible and visible to
a broad audience, thus enriching the overall media consumption experience.

  2. 2.

Fairness in Content Distribution: Central to Epistral’s mission is the
establishment of a fair and equitable system for content creators. This goal
addresses the imbalance and exploitation often experienced in traditional
media platforms, where creators’ contributions are undervalued. By leveraging
blockchain technology, Epistral ensures that content creators are justly
rewarded for their work, fostering an environment where creativity and
originality are incentivized.

  3. 3.

Sovereignty Among Users and Regulators: Sovereignty in the Epistral context
refers to the empowerment of both users and regulators within the media
ecosystem. For users, this means having control over their data and the
content they consume, free from manipulative algorithms and invasive
advertising practices. For regulators, Epistral offers a transparent and
accountable framework, enabling them to enforce regulations effectively while
respecting the principles of free expression and privacy.

Epistral Network is revolutionizing the media industry by combining media
expertise with blockchain technology. This innovative approach is ushering in
a new era of digital media that prioritizes fairness, diversity, and autonomy,
while also fostering genuine creativity and engagement. By adhering to these
principles, the network not only addresses current challenges in digital media
but also paves the way for a more inclusive, dynamic, and fair media landscape
in the future.

##  4 Core System Architecture: Attribution and Governance

The architectural design of the Epistral Network is a sophisticated
amalgamation of various blockchain technologies and principles, thoughtfully
orchestrated to address the nuanced needs of creators, consumers, and
regulators in a Hive-based blockchain environment. This multidimensional
architecture ensures that each participant group is not only integrated but
also derives optimum benefit from the network’s diverse functionalities.

Figure 2: In-depth System Components of the Epistral Network

The primary component of this ecosystem is the Blockchain Layer, which
includes the Hive Protocol and Honeycomb. This layer plays a crucial role in
creating a strong base for different parties involved. Content creators can
use this platform to register their work and keep track of royalties, taking
advantage of the unchangeable nature of blockchain that permanently records
their rights and contributions. Consumers benefit from increased transparency
and authenticity in their interactions with content, as evidenced by secure
and unalterable transaction records. Regulators find this layer essential for
monitoring compliance and promoting consensus participation, thus aligning
with legal and regulatory requirements.

The Epistral platform incorporates a Consensus Mechanism that combines DPoS
with a Double DAG system. This mechanism aims to establish a democratic
governance model where both creators and consumers actively engage in network
governance. By doing so, they contribute to a decision-making process that is
transparent and inclusive. Additionally, this mechanism provides regulators
with a transparent view of governance and network operations.

The Smart Contract Functionality in Epistral stands as a beacon of innovation.
These contracts are engineered to distribute rewards automatically,
maintaining sustainable debt-to-ownership ratios. Creators are fairly
compensated for their subjective contributions, fostering a sense of equity
and recognition. Consumers, on the other hand, engage in a trustless
environment, rewarded for their curation efforts. This setup also ensures
strict adherence to regulatory frameworks, automating compliance, and
enhancing transparency.

The incorporation of native and debt-backed EPT tokens is a crucial aspect of
Epistral’s economic model, known as tokenization. Tokenization refers to the
process of creating tokens within the network. The native EPT token is
tradable, while the EPT debt-backed token is designed to provide stability.
These tokens operate within sustainable debt-to-ownership ratios. Epistral
also utilizes the Epistral Power token for governance, which is staked by
participants. Additionally, the use of EPRC tokens for decentralized compute
resources demonstrates a comprehensive tokenization strategy. This strategy
ensures a fair and balanced distribution of tokens among stakeholders, in
accordance with Zipf’s law (Zipf, 2013)

The network’s Decentralized Storage is a vital component, which is
accomplished by integrating IPFS and utilizing RSS Syndication. This
characteristic provides creators with a strong and resistant hosting platform
that is not subject to censorship, ensuring the durability and reliability of
their content. Additionally, users benefit from improved content
accessibility, free from the risks associated with centralized hosting.
Regulators also appreciate this decentralized approach as it guarantees the
trustworthiness and genuineness of data, which is essential for maintaining
overall confidence in the system.

Cross-Chain Interoperability is another notable feature of the Epistral
Network. It facilitates the creation of accounts using other credentials,
possibly through a naming service like Ceramic light accounts or
Wire.network’s UPAP protocol. This interoperability allows access to different
chains supporting ECDSA key pairs, thereby enhancing the network’s reach and
utility. It broadens the spectrum for content distribution, allowing creators
and consumers to interact with a wider array of blockchain ecosystems.

The architecture of the Epistral Network is a prime example of innovation, as
it combines the Hive protocol with advanced features that are specifically
designed to meet the diverse requirements of its various stakeholders. This
complex design not only aligns with the network’s strategic vision, but also
represents a significant advancement in the development of digital media
platforms based on blockchain technology. Epistral aims to cater to the unique
needs of creators, consumers, and regulators, positioning itself as a
versatile and forward-looking platform in the field of blockchain.

##  5 Recommendation system: Diversity and Fairness

The Epistral Network is striving to revolutionize digital media consumption
and creation by introducing anti-mimetic principles to counter the widespread
inclination towards mimetic desire and echo chambers in social media. This
section will explore the methods used by Epistral to guarantee content
variety, impartiality, and a break from mimetic tendencies.

The implementation of antimimetic principles in the Epistral Network is a
purposeful and creative attempt to combat the homogenizing effects of
traditional social media algorithms and predict the authenticity of subjective
proof of work using cumulative reputation of stakeholders. Through the limited
lifespan of content, graph-based token production, and a recommendation
algorithm emphasizing embedding entropy, Epistral encourages a varied,
dynamic, and equitable media environment.

These mechanisms not only enhance the user experience by exposing them to a
wider array of content but also ensure that the network’s economy thrives on
the principles of diversity and originality. By doing so, Epistral sets a new
standard for digital media platforms, one that champions diversity and combats
the mimetic cycles prevalent in today’s digital landscape.

###  5.1 Content Lifecycle

The Epistral network has established a specific timeframe for the lifecycle of
its content, which is determined by network governance. This intentional
decision is made to encourage a constant stream of new and varied content. The
distribution of tokens is based on the level of engagement that the content
receives during this timeframe. This strategy encourages creators to
consistently generate engaging and thought-provoking content, thus reducing
the risk of content becoming stagnant and the formation of echo chambers.

Figure 3: Media fairness, diversity and transparency

###  5.2 Content-driven Token Economics

Epistral utilizes a graph-based approach to token generation, which evaluates
the distribution of content across its blockchain. This method assesses how
content is shared and interacted with throughout the network, focusing on
diversity and reach. The algorithm takes into account both the amount of
content and the type of transactions within the network. A higher variety of
content and a balanced transaction environment lead to increased token
generation.

This system encourages a vibrant and varied content landscape, with the
Epistral token economy being directly linked to the diversity of content and
its usage. This connection ensures that economic incentives are in line with
the network’s anti-mimetic goals, rewarding content that contributes to a
diverse and dynamic media ecosystem.

###  5.3 Entropy-based Personalisation

Epistral’s personalized recommendation algorithm is specifically designed to
ensure that the content consumed by users maintains a high level of embedding
entropy. This concept refers to the diversity and unpredictability of the
content recommendation system. While the network can establish target entropy
levels, the primary objective is to optimize for diversity and fairness in the
content feed. This approach helps to prevent users from being exposed to a
limited range of content and reduces the risk of algorithmic echo chambers.

Unlike traditional media platforms that often trap consumers in localized
content loops, Epistral’s algorithm actively opposes this trend. By
prioritizing content diversity and embedding entropy, the platform ensures
that users are not confined to narrow content silos but are instead exposed to
a wide variety of ideas and perspectives.

The recommendation system does not give equal opportunities to all content. If
a piece of content is of high quality, it will have a higher chance of being
surfaced due to the system’s design. We do not use any anti-bot system, as the
entropy-based system prevents any specific narrative from dominating the
consumer feed. Regardless of how something is curated, its value only
increases in relation to its surrounding content, not globally.

For example, if there are 100,000+ bot-produced articles about "Elon Musk"
with high votes, but only 100 articles about Albert Einstein, the "Elon Musk"
articles will not take over the user’s feed, no matter how much curation is
done. Instead, a few good articles in the "Elon Musk" neighborhood (in terms
of semantic similarity) will be shown more frequently compared to others in
that neighborhood.

##  6 Token Economy: Incentives and Rewards

The Epistral Network introduces a sophisticated token economy centered around
its native cryptocurrency, the Epistral Token (EPT). This economy is designed
to incentivize and reward the various activities within the network, aligning
the interests of creators, consumers, curators, and other stakeholders. The
token economy is fundamental to achieving the network’s goals of fairness,
diversity, and sovereignty in media consumption and creation.

Figure 4: Token economy

The EPT token is a key component of the Epistral Network’s blockchain-based
media platform, helping to create a dynamic and interactive environment. By
utilizing the token, Epistral ensures that the interests of all stakeholders
are aligned, encourages the production of high-quality content, and gives
users the opportunity to take part in the network’s governance. This token
economy not only provides a fair and rewarding system for digital media, but
also demonstrates the innovative power of blockchain technology in developing
sustainable and equitable economic models in the digital world.

#### Token Functionality and Distribution

The Epistral ecosystem utilizes the EPT token for multiple purposes: as a
medium of exchange, a measure of value, distribution of compute, management of
liquidity and a tool for governance. Users can earn EPT through activities
such as content creation, curation, and participation in network governance.
Epistral has a nuanced distribution mechanism that rewards users based on
their contribution to the network, including content creation and curation
effectiveness, community engagement, and other valuable activities. The
distribution is managed by smart contracts ensuring sustainable reducing
inflation, transparency, and fairness.

The initial distribution of tokens consists of both liquid and staked tokens,
with the distribution changing based on the debt:ownership ratios. However,
users always have the opportunity to acquire a proportional ownership stake in
the network. They can do this by either purchasing tokens through capital
contributions or by earning them through their work. It is worth noting that
reputation cannot be obtained solely through capital contributions.

#### Incentives for Creators

Creators in the Epistral network are incentivized through direct monetization
of their content. They earn EPT tokens based on community engagement metrics
like views, likes, and shares, promoting the creation of high-quality,
original content. Additionally, creators can stake their EPT tokens to earn
extra rewards and gain more influence in network governance, allowing them to
participate in important platform decisions.

#### Incentives for Consumers

Consumers are rewarded with EPT tokens for engaging with content, such as
viewing, liking, commenting, and sharing. This model fosters active
participation and enhances the network’s vibrancy. Effective curation and
discovery of trending or valuable content by consumers are also rewarded,
encouraging a community-driven approach to content promotion.

#### Governance and Staking

The Epistral network’s governance is decentralized, with token holders having
a say in its operation. The more EPT a user stakes, the greater their voting
power in decisions like feature developments and policy changes. Staking
mechanisms not only enable participation in governance but also secure the
network, with stakers receiving additional token rewards, aligning their
interests with the network’s success.

#### Token Economy Dynamics

The supply and demand of EPT are regulated to maintain its value and utility
within the ecosystem. Demand is driven by its various uses, including content
monetization, access to premium features, and governance participation. The
token economy is designed to reward the quality and uniqueness of content,
encouraging creators to produce innovative and engaging material, thus
aligning with Epistral’s anti-mimetic ethos.

#### Regulatory Compliance

Managing market fluctuations is critical to the stability of the EPT token.
Epistral implements mechanisms to mitigate extreme volatility. Additionally,
the token economy is designed with an acute awareness of regulatory
requirements, ensuring compliance with financial laws and guidelines across
various jurisdictions.

##  7 Key Innovations: Consumer, Privacy and Compliance

The Epistral Network, developed on the Hive blockchain protocol, presents a
range of innovative features and functionalities designed to revolutionize the
digital media landscape. These key aspects are tailored to meet the diverse
needs of creators, consumers, and regulators, ensuring an equitable, engaging,
and compliant media ecosystem.

###  7.1 Consumer Engagement and Interaction

For consumers, Epistral provides a personalized content discovery experience
using advanced algorithms. This feature tailors the content exposure to each
user’s preferences and interests, thereby enhancing user engagement. The
platform also incorporates interactive features, allowing consumers to engage
with content through likes, comments, and shares. These interactions not only
improve the user experience but also increase the content’s visibility and the
creator’s potential for earning rewards.

Incorporating the anti-mimetic philosophy, inspired by the works of McLuhan
and Girard, Epistral seeks to cultivate an environment where content is not
merely a reflection of existing desires and trends but a platform for genuine
expression and innovation. This approach challenges the conventional dynamics
of content creation and consumption, encouraging users to engage with content
that inspires original thought and creativity, rather than perpetuating cycles
of imitation.

###  7.2 Scalable and Secure Technology Stack

To enhance its functionality and reach, Epistral supports cross-chain
interoperability, facilitating seamless interactions with other blockchain
networks. This interoperability expands the range of content and token
exchange possibilities. Furthermore, the platform integrates oracle services
to import external data onto the blockchain. These services provide valuable
insights for content strategy and market trends, benefiting both creators and
consumers.

Epistral places a high priority on security and privacy. Leveraging Hive’s
robust security infrastructure, the platform ensures the protection of user
data and transactions. Advanced encryption and privacy-enhancing technologies
are employed to safeguard user information and the integrity of content.
Moreover, users are given control over their personal data, with options to
manage privacy settings and data sharing preferences. This approach is in line
with global privacy standards and caters to user expectations for data
sovereignty.

In terms of scalability and performance optimization, Epistral utilizes Hive’s
DPoS consensus mechanism to achieve efficient transaction processing. This
capability is essential for managing the large volumes of user interactions
and content uploads. The network’s architecture is specifically designed for
scalability, ensuring that as the platform grows, it continues to provide a
seamless and responsive experience for all users.

###  7.3 Regulatory Compliance and Governance

Epistral is committed to regulatory compliance and governance. The platform
incorporates a transparent regulatory framework to ensure adherence to various
legal standards, including content moderation, intellectual property rights
protection, and regional content regulations. Additionally, Epistral employs a
decentralized governance model, enabling stakeholders to propose and vote on
changes. This approach ensures that the network evolves in a manner that
aligns with the collective interests of its participants.

The Epistral Network’s key features and functionalities represent a
comprehensive approach to creating a decentralized, user-centric, and
regulator-friendly media platform. By addressing the needs of creators,
consumers, and regulators through innovative blockchain applications, Epistral
stands as a pioneering solution in the digital media domain, poised to reshape
how content is created, distributed, and consumed in the blockchain era.

##  8 Challenges and Limitations

The Epistral Network, a pioneering endeavor, is confronted with a variety of
difficulties and restrictions that are inherent in its design and the wider
environment in which it operates. To address these issues, a combination of
technical advancement, economic stability, strategic planning, and compliance
with regulatory requirements is necessary. It is essential for Epistral to
overcome these obstacles in order to realize its ambition of transforming the
digital media landscape through blockchain technology and anti-mimetic
principles. As the platform progresses, ongoing assessment and adjustment will
be essential for overcoming these challenges and ensuring long-term success.

#### Technical

Epistral, like many other blockchain-based systems, is confronted with the
major issue of scalability. It is essential for the network to be able to
process a large number of transactions and interactions quickly and without
any loss of speed or performance. Although Hive’s infrastructure offers
scalability advantages, Epistral must continue to develop in order to
accommodate the growing user activity and content volume. Another significant
technical challenge is the complexity of implementing and maintaining advanced
anti-mimetic algorithms. These algorithms, which are essential for
guaranteeing a wide range of content distribution and tailored
recommendations, must be able to adjust to changing user behaviors and content
trends, necessitating ongoing development and optimization.

#### Economic and Tokenization

In the realm of economics and tokenization, the volatility of the
cryptocurrency market poses a considerable challenge. Fluctuations in the
value of the EPT token can significantly impact the network’s economy,
affecting both creator incentives and user engagement. Managing this
volatility to ensure the stability of the token is a major task. Additionally,
designing a token economy that balances the incentives for creators,
consumers, and curators fairly is a delicate task. Over-incentivizing one
group could lead to imbalances and unintended network behaviors.

#### Content and User Engagement

Ensuring consistent content quality is a complex issue for Epistral. Although
the network encourages high-quality content creation, relying on community
curation and engagement metrics for content valuation could result in quality
disparities. Another major challenge is user adoption and retention.

#### Regulatory and Compliance

Navigating the complex network of regulations in different jurisdictions is a
significant challenge for Epistral, operating in a global landscape. The
platform must ensure compliance with various legal standards, especially in
areas such as data protection, content regulation, and financial transactions.
Furthermore, it is complex to balance the decentralized nature of blockchain
with regulatory compliance. Decentralization offers numerous benefits, but can
complicate the adhesion to certain regulatory frameworks that require
centralized control or oversight.

#### Limitations

Technological barriers present a limitation for less tech-savvy users engaging
with Epistral. Simplifying the user experience and educating users about
blockchain and cryptocurrencies are crucial steps to overcome this barrier.
Additionally, as a decentralized platform, Epistral has limited control over
user behavior and content. This lack of control can pose challenges in content
moderation and in ensuring a safe and respectful online community.

##  9 Case Studies and Applications

This section aims to illustrate the practical implications and possibilities
of the Epistral Network. Through hypothetical case studies and applications,
we will demonstrate how the unique features of Epistral can tackle current
issues in the digital media sphere and open up new possibilities for creators,
consumers, and regulators.

### Case Study 1: Empowering Independent Creators

  * •

Background: Jane, an independent documentary filmmaker, struggles to find a
platform that fairly compensates her for her work and offers her control over
content distribution.

  * •

Application of Epistral: On Epistral, Jane uploads her documentary. The
platform’s blockchain-based system ensures her ownership rights are protected.
Her work gains traction due to the network’s anti-mimetic content
recommendation algorithm, exposing her documentary to a diverse audience.

  * •

Outcome: Jane receives fair compensation in EPT tokens based on genuine
audience engagement within the 15-day content lifespan. She reinvests some
tokens to stake in the network, gaining a say in future governance decisions.

### Case Study 2: Enhancing Consumer Experience

  * •

Background: Alex, a content consumer, is tired of the repetitive and echo-
chamber nature of traditional social media feeds.

  * •

Application of Epistral: Alex discovers Epistral and experiences a content
feed curated based on embedding entropy, exposing him to a variety of content
genres and viewpoints.

  * •

Outcome: Alex finds the content on Epistral more enriching and less biased. He
actively participates in content curation, earning EPT tokens, which further
enhances his engagement with the platform.

### Case Study 3: Regulatory Compliance and Transparency

  * •

Background: A regulatory body seeks to ensure compliance and transparency in
digital content without imposing excessive controls.

  * •

Application of Epistral: The body utilizes Epistral’s transparent blockchain
ledger and regulatory dashboards to monitor content and transactions, ensuring
compliance with regional standards.

  * •

Outcome: The regulator effectively oversees the digital content landscape on
Epistral, ensuring it aligns with legal standards while appreciating the
platform’s decentralized governance model.

### Case Study 4: Cross-Cultural Content Collaboration

  * •

Background: A global collaboration project aims to bring together content
creators from diverse cultures to create a series of interconnected digital
art pieces.

  * •

Application of Epistral: Creators use Epistral to upload and link their
content, benefiting from the platform’s cross-chain functionality and
decentralized storage. The diverse content ecosystem of Epistral facilitates
wider exposure.

  * •

Outcome: The project achieves global recognition, with contributors receiving
equitable compensation and recognition for their work. The initiative
showcases the power of blockchain in fostering cross-cultural collaboration.

### Case Study 5: Combatting Digital Content Monopolies

  * •

Background: Small content creators face challenges competing against large
digital content monopolies.

  * •

Application of Epistral: Utilizing Epistral’s equitable token distribution and
anti-mimetic recommendation engine, these creators gain a level playing field
where quality and originality are key.

  * •

Outcome: A more diverse range of creators emerge, breaking the dominance of
larger content monopolies and enriching the digital content landscape with
unique and varied perspectives.

### Case Study 6: Transforming E-commerce Reviews through an Anti-Mimetic
Approach

  * •

Background: Emily owns a small e-commerce business that struggles to gain
visibility and credibility due to the dominance of larger retailers and the
prevalence of inauthentic reviews on traditional platforms.

  * •

Application of Epistral: Emily chooses to utilize the Epistral Network to
conduct her product evaluations. Epistral incorporates the reviews into the
blockchain, guaranteeing their credibility and openness. The platform’s anti-
mimetic strategy ensures that the reviews are varied and authentic, capturing
a broad spectrum of customer encounters.

  * •

Outcome: The anti-mimetic algorithm effectively identifies different genuine
reviews, preventing the problem of biased or uniform feedback. This openness
and variety in reviews appeal to a larger audience for Emily’s business,
improving the credibility and reliability of her brand. Customers are more
assured in their buying choices, as they know the reviews are authentic and
reflect a diverse range of opinions, not just the most popular or imitative
perspectives.

These case studies exemplify the transformative potential of the Epistral
Network in various aspects of digital media creation, consumption, and
regulation. By leveraging blockchain technology and anti-mimetic principles,
Epistral addresses key challenges faced in the current media ecosystem,
offering innovative solutions that benefit all stakeholders. As the platform
evolves, it is poised to inspire new applications and case studies, further
demonstrating its impact on the digital media world.

##  10 Future Directions

The Epistral Network is committed to staying at the forefront of blockchain
technology. To enhance scalability, security, and user experience, the network
plans to continuously integrate the latest advancements in blockchain
technology. This includes exploring layer-2 solutions, sidechains, and other
technological innovations that could significantly augment network
performance.

In the realm of content moderation and enhancement, Epistral is turning
towards artificial intelligence as a key tool for the future. AI will play a
crucial role in maintaining content quality, detecting and addressing harmful
content, and optimizing the anti-mimetic content recommendation algorithms.
The use of AI is expected to streamline these processes, ensuring a safer and
more engaging platform for users.

Epistral is also focusing on expanding its cross-chain functionalities. By
enhancing its ability to interact seamlessly with a broader range of
blockchain ecosystems, the network aims to increase its content reach,
diversify its token economy, and facilitate wider user adoption. This
expansion is seen as crucial for the network’s growth and integration within
the broader blockchain community.

Another significant area of focus for the future is user education and
community building. Epistral recognizes the importance of educating users
about blockchain technology and the specifics of its platform. Building a
strong, engaged community is seen as essential for the network’s growth and
sustainability. This approach is expected to foster a more informed and active
user base, contributing to the overall health and dynamism of the platform.

Lastly, regulatory collaboration and compliance are top priorities for
Epistral. The network aims to work closely with regulatory bodies to ensure
compliance and adapt to the evolving legal landscapes. Epistral’s goal is to
be at the forefront of defining regulatory standards for blockchain-based
media platforms, setting a benchmark for responsible and compliant operation
in this emerging field.

##  11 Conclusion

The Epistral Network represents a groundbreaking contribution to the digital
media landscape. Its innovative use of blockchain technology, combined with
anti-mimetic principles, sets it apart from traditional media platforms. Key
contributions of the Epistral Network include:

  * •

Revolutionizing Content Monetization: Epistral’s token economy and smart
contract-based monetization mechanisms offer a fair and transparent way for
content creators to earn from their work, addressing a major gap in the
current digital content market.

  * •

Promoting Content Diversity and Quality: The network’s anti-mimetic approach
to content recommendation and curation fosters a diverse and high-quality
content ecosystem. This is a significant departure from the echo chambers and
mimetic cycles prevalent in existing social media platforms.

  * •

Empowering Users and Decentralizing Governance: By giving users a stake in
network governance and rewarding participation, Epistral empowers its
community and promotes decentralized decision-making. This model is pivotal in
shaping a media platform that is truly by and for its users.

  * •

Enhancing Transparency and Regulatory Compliance: The transparent nature of
blockchain technology, as implemented in Epistral, provides a new level of
transparency and accountability in digital media. This is a major step forward
in balancing freedom of expression with regulatory compliance.

  * •

Setting a Precedent in Blockchain Media Platforms: Epistral sets a precedent
for future blockchain-based media platforms, showcasing the potential of this
technology to transform how digital content is created, distributed, and
consumed.

In conclusion, the Epistral Network not only addresses current challenges in
the digital media sphere but also opens up new possibilities for fair,
diverse, and engaging content consumption and creation. As the network
continues to evolve, it stands poised to make a lasting impact on the digital
media landscape, paving the way for more equitable, innovative, and user-
centric media platforms.

## References

  * McLuhan [1964] Marshall McLuhan.  _Understanding Media: The Extensions of Man_.  McGraw-Hill, 1964.
  * Girard [1965] René Girard.  _Deceit, Desire, and the Novel: Self and Other in Literary Structure_.  Johns Hopkins University Press, 1965.
  * Hayden [2003] Grant M Hayden.  The false promise of one person, one vote.  _Mich. L. Rev._ , 102:213, 2003.
  * Nakamoto [2008] Satoshi Nakamoto.  Bitcoin: A peer-to-peer electronic cash system, 2008.
  * Steemit Inc. [2016] Steemit Inc.  Steem whitepaper, 2016.
  * Benet [2014] Juan Benet.  Ipfs-content addressed, versioned, p2p file system.  _arXiv preprint arXiv:1407.3561_ , 2014.
  * Zipf [2013] George Kingsley Zipf.  _The psycho-biology of language: An introduction to dynamic philology_.  Routledge, 2013.


# Asymptotic rates of convergence for compact reversible Markov chains

Yue Xu  Morningside Center of Mathematics, Chinese Academy of Sciences
<EMAIL_ADDRESS>and  Xiuwu Zhu  Department of Mathematics and Yau
Mathematical Sciences Center, Tsinghua University;  Yanqi Lake Beijing
Institute of Mathematical Sciences and Applications, Beijing 101408, China.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish the main term governing the rate at which a
compact reversible Markov chain converges to stationarity. As an application,
we obtain the sharpest estimates for the corank distributions of uniform, skew
centrosymmetric, symmetric, skew-symmetric, and Hermitian matrices.

###### Key words and phrases:

reversible Markov chain, convergence rate, corank distribution, random matrix

###### 2020 Mathematics Subject Classification:

60F17, 60J10

##  1\. Introduction

Studying the convergence rate of Markov chains is an important topic in
probability theory and it has many important applications in [1, 2]. For an
irreducible, aperiodic Markov chain, the convergence rate is exponential,
which is called geometrically ergodic, if and only if it satisfies certain
drift conditions [3]. If the Markov chain is reversible, the exponential rate
is closely related to the non-one eigenvalue of the corresponding transition
matrix with largest absolute value [4]. In this paper, we view the transition
matrix as a bounded self-adjoint operator on a separable Hilbert space. If it
is further assumed that the operator is compact, we employ the spectral
decomposition theorem to obtain the main term of the convergence rate. Also,
the mixing time of this reversible Markov chain can be obtained.

The main results of this paper are to prove that the operators related to
certain classes of Markov chains are compact and to provide all of their
eigenvalues. Consequently, we can demonstrate that Markov chains related to
the corank distribution of several classes of random matrices over finite
fields are also compact, thus obtaining the asymptotic expressions for their
distributions. These corank distributions play a significant role in the
context of the Cohen-Lenstra conjecture for ideal class groups and Selmer
groups [2, 7, 8].

Let $P$ be the transition matrix of an irreducible, aperiodic Markov chain on
nonnegative integers. Assume that $P$ is reversible with respect to $\pi$.
Recall that the Hilbert space $\ell^{2}(\pi)$ consists of the complex-valued
sequences $\mu=(\mu(0),\mu(1),\cdots)$ with inner product and norm defined by

$\langle\mu,\nu\rangle_{\pi}:=\sum_{i=0}^{\infty}\frac{\mu(i)\overline{\nu(i)}}{\pi(i)},\quad\|\mu\|_{\pi}:=\left(\sum_{i=0}^{\infty}\frac{|\mu(i)|^{2}}{\pi(i)}\right)^{\frac{1}{2}}.$

Then $P$ is a bounded, self-adjoint, linear operator on $\ell^{2}(\pi)$
through the action $P\cdot\mu:=\mu P$. Recall that the modified (without the
factor $1/2$) total variation distance $\|\cdot\|_{tv}$ between
$\mu_{1},\mu_{2}\in\ell^{2}(\pi)$ is defined as follows:

$\|\mu_{1}-\mu_{2}\|_{tv}:=\sum_{i=0}^{\infty}|\mu_{1}(i)-\mu_{2}(i)|\leq\|\mu_{1}-\mu_{2}\|_{\pi}.$

######  Theorem 1.1.

Assume further that $P$ is a compact operator on $\ell^{2}(\pi)$. Let
$\lambda_{0},\lambda_{1},\lambda_{2},\cdots$ be all eigenvalues of $P$ with
non-increasing absolute value. Then for any $\mu\in\ell^{2}(\pi)$,

$\|\mu
P^{n}-\sum_{i=0}^{k}\lambda_{i}^{n}\mu_{i}\|_{tv}=O(|\lambda_{k+1}|^{n}).$

Here $\mu_{i}$ is the $\lambda_{i}$-component in the spectral decomposition of
$\mu$ and the implicit constant is less than $\|\mu\|_{\pi}$. In particular,

$\|\mu
P^{n}-\mu_{0}\|_{tv}=\begin{cases}\|\mu_{1}\|_{tv}\cdot|\lambda_{1}|^{n}+O(|\lambda_{2}|^{n}),&\mbox{if
}|\lambda_{1}|>|\lambda_{2}|\\\
(\|\mu_{1}+(-1)^{n}\mu_{2}\|_{tv})\cdot|\lambda_{1}|^{n}+O(|\lambda_{3}|^{n}),&\mbox{if
}|\lambda_{1}|=|\lambda_{2}|.\end{cases}$

Here $\mu_{0}=(\mu\cdot\mathbf{1})\pi$, $\mathbf{1}=(1,\cdots,1)^{T}$, and the
implicit constant here does not exceed
$\left(\|\mu\|_{\pi}^{2}-1\right)^{1/2}$.

To consider the corank distribution of random matrices
${\mathrm{Mat}}_{n\times(n+m)}({\mathbb{F}}_{q})$, it leads to the following
transition matrix

$P_{m}(i,j)=\begin{cases}q^{-1-2i-m},&\mbox{if }j=i+1\\\
1-(1-q^{-i})(1-q^{-m-i})-q^{-1-2i-m},&\mbox{if }j=i\\\
(1-q^{-i})(1-q^{-m-i}),&\mbox{if }j=i-1\\\ 0,&\mbox{otherwise.}\end{cases}$

It’s irreducible, aperiodic and reversible with a stationary distribution
$\pi_{m}$. We may view it as a linear operator on $\ell^{2}(\pi_{m})$.

######  Theorem 1.2.

$P_{m}$ is compact on $\ell^{2}(\pi_{m})$, and the set of eigenvalues is
${\mathrm{EV}}(P_{m}):=\left\\{q^{-k}:k\geq 0\right\\}$.

######  Remark 1.3.

In the theorem, the corresponding eigenspaces for each eigenvalues can be
constructed explicitly.

######  Remark 1.4.

In general, if $q$ and $m$ are two real numbers with $q>1$ and $m>-1$, the
theorem still holds.

######  Corollary 1.5.

Let $q$ be a prime power and $m\geq 0$ be a non-negative integer.

$\|{\mathrm{Prob}}\left(n-\mathrm{rank}~{}M=k~{}|~{}M\in{\mathrm{Mat}}_{n\times(n+m)}({\mathbb{F}}_{q})\right)-\pi_{m}\|_{tv}=\frac{2\pi_{m}(0)}{(q-1)q^{m}}\cdot\frac{1}{q^{n}}+O\left(\frac{1}{q^{2n}}\right),$

where

$\pi_{m}(i)=\frac{\eta_{\infty}(q)}{q^{i(i+m)}\eta_{i}(q)\eta_{i+m}(q)},~{}~{}\eta_{k}(q)=\prod_{i=1}^{k}(1-q^{-i})$

and the implicit constant is less than $(\pi_{m}(0)^{-2}-1)^{\frac{1}{2}}$.

For a finite abelian $p$-group $G$, denote
$\mathrm{rk}_{p}(G):=\dim_{{\mathbb{F}}_{p}}G/pG$. Following the method in [9,
Proposition 1] (or [10, Proposition 14.1]), we also obtain the cokernel
distribution of ${\mathrm{Mat}}_{n\times(n+m)}({\mathbb{Z}}_{p})$.

######  Corollary 1.6.

Let $G$ be a finite abelian $p$-group and $\mu$ be the normalized Haar measure
on ${\mathrm{Mat}}_{n\times(n+m)}({\mathbb{Z}}_{p})$, i.e.
$\mu({\mathrm{Mat}}_{n\times(n+m)}({\mathbb{Z}}_{p}))=1$. For any
$M\in{\mathrm{Mat}}_{n\times(n+m)}({\mathbb{Z}}_{p})$, define the cokernel of
$M$ by $\mathrm{coker}~{}M:={\mathbb{Z}}_{p}^{n}/M{\mathbb{Z}}_{p}^{n+m}$.
Then

$\mu\left(\left\\{M\in{\mathrm{Mat}}_{n\times(n+m)}({\mathbb{Z}}_{p})\
\big{|}\ \mathrm{coker}~{}M\simeq
G\right\\}\right)=w_{m}(G)+\lambda(G)p^{-n}+O(p^{-2n}),$

where

$w_{m}(G)=\frac{\eta_{\infty}(p)/\eta_{m}(p)}{|G|^{m}\cdot|{\mathrm{Aut}}(G)|}\text{
and }\lambda(G)=\frac{w_{m}(G)(p+p^{-m}-p^{\mathrm{rk}_{p}(G)})}{p-1}.$

######  Remark 1.7.

The distribution of ideal class groups for quadratic fields can be predicted
by cokernel model (for example, [7]). Our corollary suggests that the error
term in the corresponding Cohen-Lenstra heuristic should be on the magnitude
of $Xp^{-n}$ where $n$ represents the number of generators for class groups.
On the other hand, based on numerical results, we conjecture the error term
should be $X^{\frac{1}{2}+\frac{1}{p}}$ for $p$ odd, $X^{\frac{3}{4}}$ for
$p=2$. It suggests that $n\gg\log X$ with probability $1$ (for the upper bound
of $n$, see [16]).

######  Remark 1.8.

Both asymptotic expansion in the above two corollary can be extended to
$q^{-kn}$-term for $k\geq 2$.

We also obtain similar sharp results for the corank distributions of
symmetric, skew symmetric, skew centrosymmetric and Hermitian matrices.

##  2\. Preliminaries

Let $P$ be a transition matrix on nonnegative integers. Assume that $P$ is
irreducible and aperiodic, that $P$ has a unique stationary distribution
denoted by $\pi$. Then by the basic limit theorem, we know that for any
nonzero initial distribution $\mu$,

$\|\mu P^{n}-\pi\|_{tv}\rightarrow 0,\text{ as }n\rightarrow\infty.$

Here the modified (without the factor $1/2$) total variation distance
$\|\cdot\|_{tv}$ between two distributions $\mu_{1},\mu_{2}$ is defined as
follows:

$\|\mu_{1}-\mu_{2}\|_{tv}:=\sum_{i=0}^{\infty}|\mu_{1}(i)-\mu_{2}(i)|.$

It is natural to ask how fast $\mu P^{n}$ converges to $\pi$. In general, if
$P$ satisfies certain drift conditions [3], then the convergence rate is
exponential. Now, can we give an explicit asymptotic estimation of the
convergence rate for some $P$ ?

###  2.1. Reversible Markov chain

Assume further that $P$ is reversible with respect to $\pi$, i.e.,
$\pi(j)P(j,i)=\pi(i)P(i,j)$ for any $i,j\geq 0$. Note that $\pi(i)>0$ for all
$i$ since $P$ is irreducible. We define the Hilbert space $\ell^{2}(\pi)$ of
the complex-valued sequences by

$\ell^{2}(\pi)=\left\\{\mu=(\mu(0),\mu(1),\cdots)\in{\mathbb{C}}^{{\mathbb{Z}}_{\geq
0}}~{}\bigg{|}~{}\sum_{i=0}^{\infty}\frac{|\mu(i)|^{2}}{\pi(i)}<\infty\right\\}.$

The inner product on $\ell^{2}(\pi)$ is given by

$\langle\mu,\nu\rangle_{\pi}:=\sum_{i=0}^{\infty}\frac{\mu(i)\overline{\nu(i)}}{\pi(i)},\text{
and the norm is
}\|\mu\|_{\pi}:=\left(\sum_{i=0}^{\infty}\frac{|\mu(i)|^{2}}{\pi(i)}\right)^{\frac{1}{2}}.$

By the Cauchy-Schwarz inequality, we have $\|\mu\|_{tv}\leq\|\mu\|_{\pi}$. In
fact,

$\|\mu\|_{tv}^{2}=\left(\sum_{i=0}^{\infty}\frac{|\mu(i)|}{\sqrt{\pi(i)}}\cdot\sqrt{\pi(i)}\right)^{2}\leq\|\mu\|_{\pi}^{2}.$

There is a natural action of $P$ on $\ell^{2}(\pi)$ through $P\cdot\mu:=\mu
P$.

Usually, the following spaces are more frequently used literature (for
example, [3]). For $1\leq p\leq\infty$, consider

$\ell^{p}_{\mathrm{old}}(\pi):=\left\\{f=(f(0),f(1),\cdots)^{T}\in{\mathbb{C}}^{{\mathbb{Z}}_{\geq
0}}~{}\bigg{|}~{}\|f\|_{\ell^{p}}<\infty\right\\},$

and

$P\cdot
f:=Pf,\quad\|P\|_{\ell^{p}}:=\sup_{\|f\|_{\ell^{p}}=1}\|Pf\|_{\ell^{p}}.$

Here,
$\|f\|_{\ell^{p}}:=\left(\sum_{i=0}^{\infty}|f(i)|^{p}\pi(i)\right)^{1/p}$ for
$p\neq\infty$ and $\|f\|_{\ell^{\infty}}:=\sup_{i}|f(i)|$. Note that both
$\|P\|_{\ell^{1}}$ and $\|P\|_{\ell^{\infty}}$ are no greater than one. Then
by Hölder’s inequality, we have $\|P\|_{\ell^{2}}\leq 1$. Meanwhile,
$\ell^{2}_{\mathrm{old}}(\pi)$ is a Hilbert space where the inner product is
given by $\langle
f,g\rangle_{\ell^{2}}:=\sum_{i=0}^{\infty}f(i)\overline{g(i)}\pi(i)$. In
particular, we have the following isomorphism between two Hilbert spaces which
is also compatible with the action of $P$.

$\phi:\ell^{2}(\pi)\rightarrow\ell^{2}_{\mathrm{old}}(\pi),\quad\mu\mapsto(\cdots,\mu(i)/\pi(i),\cdots)^{T}.$

Then

$\|P\|_{\pi}:=\sup_{\|\mu\|_{\pi}=1}\|\mu P\|_{\pi}=\|P\|_{\ell^{2}}\leq 1.$

In other word, $P$ is a linear contraction on $\ell^{2}(\pi)$.

On the other hand, the adjoint operator $P^{\ast}$ on $\ell^{2}(\pi)$ is
defined by

$P^{\ast}(i,j):=\frac{P(j,i)\pi(j)}{\pi(i)}.$

Hence, as a operator, $P$ is also self-adjoint since $P^{\ast}=P$. For similar
reason, we have an isomorphism between $\ell^{2}(\pi)$ and

$\ell^{2}:=\left\\{\nu=(\nu(0),\nu(1),\cdots)^{T}\in{\mathbb{C}}^{{\mathbb{Z}}_{\geq
0}}~{}\bigg{|}~{}\|\nu\|:=\left(\sum_{i=0}^{\infty}|\nu(i)|^{2}\right)^{1/2}<\infty\right\\}$

by sending $\mu$ to $\mu/\sqrt{\pi}$. Hence $\ell^{2}(\pi)$ is separable. In
conclusion, $P$ is a bounded, self-adjoint, linear operator on the separable
space $\ell^{2}(\pi)$.

###  2.2. Spectral theory

######  Definition 2.1 (The spectrum of a linear operator).

Let $T$ be a linear operator on a complex Hilbert space $X$. The spectrum
denoted by $\sigma(T)$ is defined by

$\sigma(T):=\\{\lambda\in{\mathbb{C}}:(\lambda I-T)\text{ is not bijective
}\\}.$

The spectrum $T$ is divided into $3$ disjoint subsets.

  1. (a)

The point spectrum, or the set of all eigenvalues of $T$, is defined by

$\sigma_{p}(T)={\mathrm{EV}}(T)=\\{\lambda\in\sigma(T):{\mathrm{Ker}}(\lambda
I-T)\neq 0\\}.$

  2. (b)

The continuous spectrum of $T$ is the set defined by

$\sigma_{c}(T)=\\{\lambda\in\sigma(T):{\mathrm{Ker}}(\lambda I-T)=0,\text{ and
}\overline{{\mathrm{Im}}(\lambda I-T)}=X\\}.$

  3. (c)

The residual spectrum of $T$ is the set defined by

$\sigma_{r}(T)=\\{\lambda\in\sigma(T):{\mathrm{Ker}}(\lambda I-T)=0\text{ and
}\overline{{\mathrm{Im}}(\lambda I-T)}\subsetneq X\\}.$

######  Theorem 2.2 (Spectral theorem).

Let $T$ be a bounded self-adjoint linear operator on an infinite dimensional
separable complex Hilbert space $X$. Then

  1. $(1)$

$\sigma(T)$ is a closed subset in $B(0,\|T\|)$;

  2. $(2)$

$\sigma_{r}(T)=\emptyset$;

  3. $(3)$

all the eigenvalues are real;

  4. $(4)$

eigenvectors associated with different eigenvalues are orthogonal.

Assume further that $T$ is compact, then

  1. $(5)$

all eigenspaces are finite dimensional;

  2. $(6)$

For any $r>0$, there are only finitely many eigenvalues of $T$ with absolute
value greater than $r$.

  3. $(7)$

$\sigma(T)=\\{0\\}\cup\sigma_{p}(T)$ and at least one of $-\|T\|,\|T\|$ is an
eigenvalue of $T$;

  4. $(8)$

Arrange all eigenvalues by their absolute value:
$\|T\|=|\lambda_{0}|\geq|\lambda_{1}|\geq|\lambda_{2}|\geq\cdots$. Then

$\ell^{2}(\pi)=\overline{\bigoplus_{i\geq 0}V_{\lambda_{i}}},$

where $V_{\lambda_{i}}$ are eigenspaces associated with $\lambda_{i}$.

For the completeness, we add some details of proof (see [11, 12]).

###### Proof.

  1. (1)

For $|\lambda|>\|T\|$, $\lambda I-T$ is invertible with

$\left(I-\frac{T}{\lambda}\right)^{-1}=\sum_{k=0}^{\infty}\left(\frac{T}{\lambda}\right)^{k}.$

If $\lambda I-T$ is invertible, for any $\gamma\in B(\lambda,\|\lambda
I-T\|)$, $\gamma I-T$ is also invertible since

$\gamma I-T=(\lambda I-T)(I+(\gamma-\lambda)(\lambda I-T)^{-1}).$

  2. (2)

See [12, Theorem 9.2-4].

  3. (3)

For non-zero element $x\in V_{\lambda}$, we have $\lambda\langle
x,x\rangle=\langle Tx,x\rangle=\langle x,Tx\rangle=\overline{\lambda}\langle
x,x\rangle$.

  4. (4)

For non-zero elements $x\in V_{\lambda}$ and $y\in V_{\gamma}$, we have
$\lambda\langle x,y\rangle=\gamma\langle x,y\rangle$.

  5. (5)

$V_{\lambda}$ is a closed subspace of $X$, hence a Hilbert space. Since
$T/\lambda=I$ is compact on $V_{\lambda}$, so is $\|\cdot\|$. Then the unit
ball in $V_{\lambda}$ is compact, hence $\dim V_{\lambda}<\infty$.

  6. (6)

Suppose there are infinite eigenvalues $\lambda_{i}$ with $|\lambda_{i}|\geq
r>0$. For each $\lambda_{j}$, take $x_{j}\in V_{\lambda_{j}}$ with
$\|x_{j}\|=1$. Since
$\|Tx_{i}-Tx_{j}\|=\sqrt{\lambda_{i}^{2}+\lambda_{j}^{2}}>r$, $\\{Tx_{i}\\}$
has no Cauchy subsequence. It contradict to the compactness of $T$.

  7. (7)

Recall the Fredholm alternative: suppose that $T:X\rightarrow X$ is a compact
operator on a Banach space $X$, and $\lambda\neq 0$. Then $\lambda I-T$ is
surjective if and only if it is injective. In particular, if $\lambda$ is not
an eigenvalue of $T$, and $\lambda\neq 0$, then $\lambda I-T$ is an
isomorphism. On the other hand, if $0\notin\sigma(T)$, then $T,T^{-1}$, and
$I=TT^{-1}$ are compact. This contradict to the assumption that $X$ is
infinite dimensional. Recall that,

$\|T\|=\sup_{\|x\|=1}\langle Tx,x\rangle\text{ or }-\inf_{\|x\|=1}\langle
Tx,x\rangle.$

In the first case, there is a sequence $x_{n}$, all with norm $1$, such that
$\langle Tx,x\rangle\rightarrow\|T\|$, and then $Tx_{n}-\|T\|x_{n}\rightarrow
0$. There must be a limit point $x$ such that $Tx=\|T\|x$, i.e. $\|T\|$ is an
eigenvalue of $T$. The second case deduce that $-\|T\|$ is an eigenvalue of
$T$.

  8. (8)

We shall prove that $W:=\left(\oplus_{i\geq
0}V_{\lambda_{i}}\right)^{\perp}=\\{0\\}$. If not, then
$\sigma(T|_{W})=\\{0\\}$ without any eigenvalues. Hence $T|_{W}=0$, i.e.,
$\\{0\\}\neq W\subset{\mathrm{Ker}}~{}T$ which means that $0$ is an
eigenvalue. Then it contradict to $W\perp{\mathrm{Ker}}~{}T$.

∎

Here we present the proof of Theorem 1.1, this theorem is a generalization of
fact 3 in [1].

###### Proof of theorem 1.1.

Since $P$ is irreducible, there is a unique stationary distribution $\pi$. It
is an eigenvector associated with the eigenvalue $1$. Note that for any
$\mu\in V_{\lambda},\lambda\neq 1$, $\mu\mathbf{1}=0$ since $\mu
P\mathbf{1}=\mu\mathbf{1}=\lambda\mu\mathbf{1}$. Now we prove that
$\lambda_{1}\neq-1$. If not, we take a nonzero eigenvector
$\mu=\mu_{+}-\mu_{-}$ where $\mu_{\pm}>0$. Without loss of generality, we
assume that $\sum_{i=0}^{\infty}\mu_{+}(i)=1=\sum_{i=0}^{\infty}\mu_{-}(i)$
since $\sum_{i=0}^{\infty}\mu(i)=0$. Then by the basic limit theorem,
$\mu=(-1)^{2n}\mu=\mu P^{2n}=\mu_{+}P^{2n}-\mu_{-}P^{2n}\rightarrow\pi-\pi=0$
as $n\rightarrow\infty$. Similarly, we know that $V_{1}=\langle\pi\rangle$.

Using the spectral theorem, for any $\mu\in\ell^{2}(\pi)$, we write
$\mu=\sum_{i\geq 0}\mu_{i},\mu_{i}\in V_{\lambda_{i}}$. Note that
$\mu_{0}=(\mu\cdot\mathbf{1})\pi$. Now we have

$\|\mu
P^{n}-\sum_{i=0}^{k}\lambda_{i}^{n}\mu_{i}\|_{\pi}^{2}=\sum_{i=k+1}^{\infty}\lambda_{i}^{2n}\|\mu_{i}\|_{\pi}^{2}=\lambda_{k+1}^{2n}\cdot\left(\sum_{i=k+1}^{\infty}\|\mu_{i}\|_{\pi}^{2}\cdot\left(\frac{\lambda_{i}}{\lambda_{k+1}}\right)^{2n}\right)\leq(\|\mu\|_{\pi}^{2}-\sum_{i=0}^{k}\|\mu_{i}\|_{\pi}^{2})\cdot\lambda_{k+1}^{2n}.$

Thus

$\|\mu
P^{n}-\sum_{i=0}^{k}\lambda_{i}^{n}\mu_{i}\|_{tv}=O(|\lambda_{k+1}|^{n}).$

In particular, if $|\lambda_{1}|>|\lambda_{2}|$, we have

$\|\mu P^{n}-\pi-\lambda_{1}^{n}\mu_{1}\|_{tv}=O(|\lambda_{2}|^{n}).$

Then

$\|\mu
P^{n}-\pi\|_{tv}=\|\mu_{1}\|_{tv}\cdot|\lambda_{1}|^{n}+O(|\lambda_{2}|^{n}).$

If $|\lambda_{1}|=|\lambda_{2}|$, we have

$\|\mu
P^{n}-\mu_{0}\|_{tv}=(\|\mu_{1}+(-1)^{n}\mu_{2}\|_{tv})\cdot|\lambda_{1}|^{n}+O(|\lambda_{3}|^{n}),\text{
as }n\rightarrow\infty.$

∎

##  3\. Hilbert-Schmidt Markov chains

######  Definition 3.1.

A bounded linear operator $T$ on a separable Hilbert space $X$ is Hilbert-
Schmidt if there is an orthonormal basis $\\{e_{n}:n\geq 0\\}$ such that
$\|T\|_{\mathcal{HS}}:=\left(\sum_{i=0}^{\infty}\|Te_{n}\|^{2}\right)^{\frac{1}{2}}<\infty$.

Hilbert-Schmidt operator is compact. In fact, let $P_{N}$ be the projection
onto the finite dimension space spanned by $\\{e_{1},\cdots,e_{N}\\}$. Then
$P_{N}T$ is compact (finite rank operator) and converge to $T$ uniformly.

Let $\\{\mu_{j}\\}$ be an orthonormal basis consist of eigenvectors. Note that

$1=\|\mu_{j}\|^{2}=\sum_{n=0}^{\infty}\langle e_{n},\mu_{j}\rangle^{2}\text{
and }1=\|e_{k}\|^{2}=\sum_{j}\langle e_{k},\mu_{j}\rangle^{2}.$

On the other hand, denoting $d_{i}=\dim V_{\lambda_{i}}$, then

$\sum_{n=0}^{\infty}\|Te_{n}\|^{2}=\sum_{n=0}^{\infty}\sum_{j}\lambda_{j}^{2}\langle
e_{n},\mu_{j}\rangle^{2}=\sum_{i}^{\infty}d_{i}\lambda_{i}^{2},$

So $T$ is Hilbert-Schmidt if and only if $T$ is compact and
$\sum_{i=0}^{\infty}d_{i}\lambda_{i}^{2}<\infty$. In particular,
$\|T\|_{\mathcal{HS}}^{2}=\sum_{i=0}^{\infty}d_{i}\lambda_{i}^{2}$.

######  Definition 3.2.

Let $P$ be an irreducible, aperiodic, and reversible transition matrix with
respect to $\pi$. Then $P$ is called Hilbert-Schmidt if it is Hilbert-Schmidt
as an operator on $\ell^{2}(\pi)$.

For any two real numbers $q>1$ and $m>-1$, consider the following transition
matrix

$P_{m}(i,j)=\begin{cases}q^{-1-2i-m},&\mbox{if }j=i+1\\\
1-(1-q^{-i})(1-q^{-m-i})-q^{-1-2i-m},&\mbox{if }j=i\\\
(1-q^{-i})(1-q^{-m-i}),&\mbox{if }j=i-1\\\ 0,&\mbox{otherwise.}\end{cases}$

It is irreducible and aperiodic since $P_{m}(i,i)>0$ for all $i\geq 0$. Define
$\pi_{m}$ by

$\pi_{m}(i)=\frac{\theta_{m}(q)}{q^{i(i+m)}\eta_{i}(q)\prod_{j=1}^{i}(1-q^{-m-j})},\text{
where }\eta_{k}(q)=\prod_{i=1}^{k}(1-q^{-i}),$

and

$\theta_{m}(q)^{-1}:=\sum_{i=0}^{\infty}\frac{1}{q^{i(i+m)}\eta_{i}(q)\prod_{j=1}^{i}(1-q^{-m-j})}<\infty$

Then $P_{m}$ is also reversible with respect to $\pi_{m}$. Now we prove that
$P_{m}$ is Hilbert-Schmidt.

###### Proof of theorem 1.2.

Firstly, we prove that for any $k\geq 1$, $q^{-k}$ is an eigenvalue. Define
$\pi_{m}\circ q^{i}\in{\mathbb{C}}^{{\mathbb{Z}}_{\geq}0}$ as follows.

$(\pi_{m}\circ q^{i})(k):=\pi_{m}(k)\cdot q^{ik}.$

It’s easy to check that $\pi_{m}\circ q^{i}\in\ell^{2}(\pi_{m})$. Now, we
claim that there exists some $a_{0},\cdots,a_{k}\in{\mathbb{R}}$ such that
$\sum_{i=0}^{k}a_{i}\cdot(\pi_{m}\circ q^{i})$ is a eigenvector associated
with $q^{-k}$. If so, for any $l\geq 0$, it must have

$\left(\sum_{i=0}^{k}a_{i}\cdot(\pi_{m}\circ
q^{i})P_{m}\right)(l)=q^{-k}\left(\sum_{i=0}^{k}a_{i}\cdot\pi_{m}(l)\cdot
q^{il}\right)$

By the reversibility, it is equivalent to

$\sum_{j=l-1}^{l+1}\left(\sum_{i=0}^{i}a_{i}q^{ij}\right)P_{m}(l,j)=q^{-k}\left(\sum_{i=0}^{k}a_{i}q^{il}\right)$

Since $\sum_{j=l-1}^{l+1}P_{m}(l,j)=1$, we have

$P_{m}(l,l-1)\sum_{i=0}^{k}a_{i}(q^{-i}-1)q^{il}+P_{m}(l,l+1)\sum_{i=0}^{k}a_{i}(q^{i}-1)q^{il}=\left(q^{-k}-1\right)\left(\sum_{i=0}^{k}a_{i}q^{il}\right),$

i.e.,

$\left(1-(1+q^{-m})q^{-l}+q^{-m-2l}\right)\sum_{i=0}^{k}a_{i}(q^{-i}-1)q^{il}+q^{-1-m-2l}\sum_{i=0}^{k}a_{i}(q^{i}-1)q^{il}=\left(q^{-k}-1\right)\left(\sum_{i=0}^{k}a_{i}q^{il}\right)$

Compare the coefficient of $q^{il}$ in two sides and denote
$a_{k+1}=a_{k+2}=0$ formally. Then

$(q^{-i}-q^{-k})a_{i}-(1+q^{-m})(q^{-1-i}-1)a_{i+1}+(q^{-2-i-m}-q^{-m}+q^{i+1-m}-q^{-1-m})a_{i+2}=0,~{}0\leq
i\leq k.$

Hence, there are unique solution of $\\{a_{0},\cdots,a_{k}\\}$ up to scalar.

Now we will prove these eigenvectors can generate whole space
$\ell^{2}(\pi_{m})$.

One can find that $vP_{m}=\lambda v$ has only one solution up to scalar, i.e.
all $V_{\lambda}$ are $1$-dimensional. Then the $P$-invariant subspace

$V:=\langle\pi\circ q^{i},i\geq 0\rangle=\overline{\bigoplus_{i\geq
0}V_{q^{-i}}}.$

Now it is sufficient to show that $\delta_{0}\in V$, where for any $i\geq 0$,
$\delta_{i}\in\ell^{2}(\pi_{m})$ is given by $\delta_{i}(k)=1$ if $k=i$ and
$0$ otherwise. In fact, if $\delta_{0}\in V$, $\delta_{1}$ will belong to $V$
since it’s a linear combination of $\delta_{0}$ and $\delta_{0}P$. By
induction argument, all $\delta_{i}$, generators of $\ell^{2}(\pi_{m})$, lie
in $V$. From the identity of $q$-series (due to Euler [13, eq(19)]), we have

$\sum_{i=0}^{\infty}b_{i}t^{i}:=\sum_{i=0}^{\infty}\frac{(-1)^{i}}{\prod_{j=1}^{i}(q^{j}-1)}t^{i}=\prod_{k=1}^{\infty}\left(1-q^{-k}t\right).$

Take $c_{k}=\frac{b_{k}}{\eta_{\infty}(q)\pi_{m}(0)}$. We claim that

$\lim_{N\rightarrow\infty}\sum_{k=0}^{N}c_{k}(\pi_{0}\circ
q^{k})=\delta_{0}\in\ell^{2}(\pi_{m}).$

i.e.,

$\lim_{N\rightarrow\infty}\pi_{m}(0)\cdot\left|\sum_{k=0}^{N}c_{k}-\pi_{m}(0)^{-1}\right|^{2}+\frac{1}{\eta_{\infty}(q)^{2}\pi_{m}(0)^{2}}\cdot\sum_{i=1}^{\infty}\pi_{m}(i)\cdot\left|\sum_{k=0}^{N}b_{k}q^{ki}\right|^{2}=0.$

By the definition of $c_{k}$, the first summand converge to $0$. It suffices
to show

$\lim_{N\rightarrow\infty}\sum_{i=1}^{\infty}\frac{1}{q^{i^{2}+mi}}\cdot\left|\sum_{k=0}^{N}b_{k}q^{ki}\right|^{2}=0.$

Since $\sum_{k=0}^{\infty}b_{k}q^{ki}=0$ ,we have

$\left|\sum_{k=0}^{N}b_{k}q^{ki}\right|=\left|\sum_{k=N+1}^{\infty}b_{k}q^{ki}\right|.$

Note that

$\left|\frac{b_{k}q^{ki}}{b_{k-1}q^{(k-1)i}}\right|=\frac{q^{i}}{q^{k}-1}.$

According to the property of alternating series, we have

$\begin{split}\sum_{i=1}^{N}\frac{1}{q^{i^{2}+mi}}\cdot\left|\sum_{k=N+1}^{\infty}b_{k}q^{ki}\right|^{2}&\leq\sum_{i=1}^{N}\frac{1}{q^{i^{2}+mi}}b_{N+1}^{2}q^{2(N+1)i}\\\
&\leq\frac{1}{\eta_{\infty}(q)^{2}}\frac{1}{q^{(N+1)(N+2)}}\sum_{i=1}^{N}\frac{1}{q^{i^{2}+mi}}q^{2(N+1)i}\\\
&\ll\frac{1}{q^{(N+1)(N+2)}}\sum_{i=1}^{N+1}q^{2(N+1)i-mi-i^{2}}.\end{split}$

Define $F(N):=\sum_{i=0}^{\infty}q^{2Ni-mi-i^{2}}$ and
$f(N)=\frac{F(N)}{q^{N(N+1)}}$. Then

$F(N+1)=1+q^{2N+1-m}\sum_{i=1}^{\infty}q^{2N(i-1)-m(i-1)-(i-1)^{2}}=1+q^{2N+1-m}F(N).$

Hence,

$f(N+1)=\frac{1}{q^{(N+1)(N+2)}}+\frac{f(N)}{q^{m+1}}$

and $f(N)\rightarrow 0$ as $N\rightarrow\infty$.

On the other hand,

$\begin{split}\sum_{i=N+1}^{\infty}\frac{1}{q^{i^{2}+mi}}\cdot\left|\sum_{k=0}^{N}b_{k}q^{ki}\right|^{2}&\leq\sum_{i=N+1}^{\infty}\frac{1}{q^{i^{2}+mi}}b_{N}^{2}q^{2Ni}\\\
&\ll\sum_{i=N+1}^{\infty}q^{-i^{2}-mi+2Ni-N(N+1)}\\\
&\ll\frac{1}{q^{(m+1)N}}\sum_{i=1}^{\infty}q^{-(i+m/2)^{2}}\rightarrow 0\text{
as }N\rightarrow\infty.\end{split}$

Hence $\\{\pi\circ q^{i}|i\geq 0\\}$ form an orthogonal basis of
$\ell^{2}(\pi_{m})$ and
$\|P_{m}\|_{\mathcal{HS}}^{2}=\sum_{i=0}^{\infty}q^{-2k}=(1-q^{-2})^{-1}$.
Therefore $P_{m}$ is Hilbert-Schmidt. ∎

######  Remark 3.3.

From a numerical perspective, we can study eigenvalues using the truncation
method [6].

######  Lemma 3.4.

Let $m>-1$ be a real number. If $f(z)=\sum_{i=0}^{\infty}\mu_{i}z^{i}$
satisfying $\sum_{i=0}^{\infty}\mu_{i}^{2}q^{i^{2}+mi}<\infty$ and for any
$k\geq 0$, $f(q^{k})=0$, then $f=0$.

###### Proof.

Let $\mu=(\cdots,\mu_{i},\cdots)$. Note that

$\frac{1}{q^{k(k+m)}}\ll\pi_{m}(k)\ll\frac{1}{q^{k(k+m)}}.$

So $\sum_{i=0}^{\infty}\mu_{i}^{2}q^{i^{2}+im}<\infty$ if and only if
$\mu\in\ell^{2}(\pi_{m})$. On the other hand, $f(q^{k})=0,\forall k\geq 0$ is
equivalent to $\langle\mu,\pi_{m}\circ q^{i}\rangle_{\pi_{m}}=0$ for all
$i\geq 0$. Hence,

$\mu\in\langle\pi_{m}\circ q^{i}|i\geq
0\rangle^{\perp}=\ell^{2}(\pi_{m})^{\perp}=\\{0\\}.$

∎

######  Remark 3.5.

  1. (1)

It is equivalent to say that the infinite Vandermonde matrix
$\mathrm{Van}(1,q,q^{2},q^{3},\cdots)$ is injective on $\ell^{2}(\pi_{m})$.

  2. (2)

Unlike the conclusion of Carlson’s theorem [5] in complex analysis, the main
difficulties here are that we cannot control the growth of $f(z)$ and $q^{k}$
are too sparse.

  3. (3)

If $m<-1$, the lemma is not true. Recall that

$\sum_{i=0}^{\infty}b_{i}q^{i}\cdot
t^{i}=\sum_{i=0}^{\infty}\frac{(-1)^{i}q^{i}}{\prod_{j=1}^{i}(q^{j}-1)}t^{i}=\prod_{k=0}^{\infty}\left(1-q^{-k}t\right).$

Take $\mu_{i}=b_{i}q^{i}$. Then $f\neq 0$ but we always have
$\sum_{i=0}^{\infty}\mu_{i}^{2}q^{i^{2}+im}<\infty$ for any $m<-1$.

##  4\. Corank distribution of random matrices

In this section, we will investigate specific Markov chains that arise from
the corank distributions of different types of matrices over finite fields.
These problems have been extensively studied in many papers (see [2, 14]).
After proving that these Markov chains are all Hilbert-Schmidt, we can deduce
the asymptotic expressions for the corank distributions of these matrices from
the earlier theorem.

###  4.1. Uniform case

Let $q$ be a prime power and $m$ a nonnegative integer. The first example we
consider is chosen uniformly from all $n\times(n+m)$ matrices over a finite
field ${\mathbb{F}}_{q}$. Define $\mathrm{corank}~{}M=n-\mathrm{rank}~{}M$.
From [14, Section 1], we know that

${\mathrm{Prob}}\left(\mathrm{corank}~{}M=k~{}|~{}M\in{\mathrm{Mat}}_{n\times(n+m)}({\mathbb{F}}_{q})\right)=(\delta_{0}P_{m}^{n})(k),$

and that

$\pi_{m}(i)=\frac{\eta_{\infty}(q)}{q^{i(i+m)}\eta_{i}(q)\eta_{i+m}(q)}.$

(one can verify that this $\pi_{m}$ is the same as the one defined in section
3). According to theorems 1.2 and 1.1, we get the corollary 1.5.

###### Proof of corollary 1.5.

By the theorem 1.2, we know that the first non-one eigenvalue of $P_{m}$ is
$q^{-1}$, and

$\nu:=\pi_{m}-\frac{\pi_{m}\circ q}{1+q^{-m}}$

is an associated eigenvector (unique up to scalar since $V_{q^{-1}}$ is
$1$-dimensional). Recall that the $k$-moment of $\pi_{m}$ is defined by
$M(\pi_{m},k):=\sum_{i=0}^{\infty}\pi_{m}(i)q^{ki}$. It is known that (see
[15, Example 6.6])

$M(\pi_{m},0)=1,M(\pi_{m},1)=1+q^{-m},M(\pi_{m},2)=1+(q+1)q^{-m}+q^{-2m}.$

Then we have

$\langle\nu,\nu\rangle_{\pi_{m}}=\sum_{i=0}^{\infty}\pi_{m}(i)-\frac{2}{1+q^{-m}}\sum_{i=0}^{\infty}\pi_{m}(i)q^{i}+\frac{1}{(1+q^{-m})^{2}}\sum_{i=0}^{\infty}\pi_{m}(i)q^{2i}=\frac{(q-1)q^{m}}{(q^{m}+1)^{2}}$

Thus the $q^{-1}$-component of $\delta_{0}$ is

$(\delta_{0})_{q^{-1}}=\frac{\langle\delta_{0},\nu\rangle_{\pi_{m}}}{\langle\nu,\nu\rangle_{\pi_{m}}}\nu=\frac{(q^{m}+1)^{-1}}{\langle\nu,\nu\rangle_{\pi_{m}}}\nu=\frac{q^{m}+1}{(q-1)q^{m}}\nu.$

On the other hand, note that $\nu(0)>0$ and $\nu(i)\leq 0$ for any $i>0$. So
$\|\nu\|_{tv}=\frac{2\pi_{m}(0)}{q^{m}+1}$ since
$\sum_{i=0}^{\infty}\nu(i)=0$. Therefore

$\|(\delta_{0})_{q^{-1}}\|_{tv}=\frac{2\pi_{m}(0)}{(q-1)q^{m}}$

∎

######  Remark 4.1.

  1. (1)

In [2], Jason Fulman and Larry Goldstein prove that (note that
$\|\cdot\|_{tv}=2\|\cdot\|_{TV}$.)

$\frac{1}{4q^{n+m+1}}\leq\|\delta_{0}P_{m}^{n}-\pi_{m}\|_{tv}\leq\frac{6}{q^{n+m+1}}.$

  2. (2)

Note that $M(\pi_{m},k)=(\pi_{m}\circ q^{k})\mathbf{1}$ and
$a_{0}\pi_{m}+a_{1}(\pi_{m}\circ q)\in V_{q^{-1}}$ for some
$a_{0},a_{1}\in{\mathbb{R}}$. According to the fact that
$V_{\lambda}\perp\mathbf{1}$ if $\lambda\neq 1$, we can get $M(\pi_{m},1)$ and
by induction, $M(\pi_{m},k)$.

This distribution is closely related to the model of Cohen-Lenstra heuristic
for ideal class groups [7].

###### Proof of corollary 1.6.

Recall that the $p^{i}$-rank of $G$ is given by

$\mathrm{rk}_{p^{i}}G=\dim_{{\mathbb{F}}_{p}}p^{i-1}G/p^{i}G.$

By [9, Proposition 1], or [10, Proposition 14.1] (they prove the case $m=0$
and $1$ and the proof of general situation is the same),

$\begin{split}&\mu\left(\left\\{M\in{\mathrm{Mat}}_{n\times(n+m)}({\mathbb{Z}}_{p})\
\big{|}\ \mathrm{coker}~{}M\simeq G\right\\}\right)\\\
&=\frac{w_{m}(G)}{\pi_{m}(\mathrm{rk}_{p}(G))}\cdot{\mathrm{Prob}}(\mathrm{corank}~{}\overline{M}=\mathrm{rk}_{p}(G)~{}|~{}\overline{M}\in{\mathrm{Mat}}_{n\times(n+m)}({\mathbb{F}}_{p}))\\\
&=\frac{w_{m}(G)}{\pi_{m}(\mathrm{rk}_{p}(G))}\cdot\left(\pi_{m}(\mathrm{rk}_{p}(G))+\frac{p^{m}+1}{(p-1)p^{m}}\nu(\mathrm{rk}_{p}(G))+O(p^{-2n})\right).\end{split}$

∎

###  4.2. Skew centrosymmetric case

Here we assume $q$ is odd. Consider the skew centrosymmetric matrices, i.e.,
${\mathrm{Mat}}^{\mathrm{scs}}_{n}({\mathbb{F}}_{q}):=\left\\{M\in{\mathrm{Mat}}_{n}({\mathbb{F}}_{q}):M_{ij}=-M_{ji}=M_{n+1-j,n+1-i}\right\\}$.
It is known that the rank is even (see [2]). Actually, we have

${\mathrm{Prob}}(\mathrm{corank}~{}M=2k|M\in{\mathrm{Mat}}_{2n}^{\mathrm{scs}}({\mathbb{F}}_{q}))={\mathrm{Prob}}(\mathrm{corank}~{}M=k|M\in{\mathrm{Mat}}_{n}({\mathbb{F}}_{q}))$
${\mathrm{Prob}}(\mathrm{corank}~{}M=2k+1|M\in{\mathrm{Mat}}_{2n+1}^{\mathrm{scs}}({\mathbb{F}}_{q}))={\mathrm{Prob}}(\mathrm{corank}~{}M=k|M\in{\mathrm{Mat}}_{n\times(n+1)}({\mathbb{F}}_{q}))$

Therefore, this corank distribution follows from the uniform case.

###  4.3. Symmetric case

Let $q$ be a prime power. Now we discuss symmetric matrices, i.e.,
${\mathrm{Mat}}_{n}^{\mathrm{sym}}({\mathbb{F}}_{q}):=\left\\{M\in{\mathrm{Mat}}_{n}({\mathbb{F}}_{q}):M^{T}=M\right\\}$.
Similarly, from [14], we have

${\mathrm{Prob}}\left(\mathrm{corank}~{}M=k~{}|~{}M\in{\mathrm{Mat}}_{n}^{\mathrm{sym}}({\mathbb{F}}_{q})\right)=(\delta_{0}P_{\mathrm{sym}}^{n})(k).$

Here

$P_{\mathrm{sym}}(i,j)=\begin{cases}q^{-i-1},&\mbox{if }j=i+1\\\
q^{-i}-q^{-i-1},&\mbox{if }j=i\\\ 1-q^{-i},&\mbox{if }j=i-1\\\
0,&\mbox{otherwise.}\end{cases}$

and

$\pi_{\mathrm{sym}}(n)=\frac{\alpha(q)}{\prod_{i=1}^{n}(q^{i}-1)},\text{ where
}\alpha(q)=\prod_{i=1,\text{odd}}^{\infty}(1-q^{-i}).$

######  Theorem 4.2.

$P_{\mathrm{sym}}$ is Hilbert-Schmidt on $\ell^{2}(\pi_{\mathrm{sym}})$ and
${\mathrm{EV}}(P_{\mathrm{sym}})=\left\\{\pm q^{-k}:k\geq 0\right\\}$.

###### Proof.

Similarly, the eigenvectors associated with eigenvalue $\pm q^{-k}$ can be
obtained as a linear combination of vectors $\\{\pi_{m}\circ(\pm
q^{i}):i=0,\cdots,k\\}$. Now it remains to prove that if
$\mu\in\ell^{2}(\pi_{\mathrm{sym}})$ (i.e.,
$\sum_{i=0}^{\infty}\mu(i)^{2}q^{\frac{i^{2}}{2}}<\infty$) such that
$\mu\perp(\pi_{\mathrm{sym}}\circ(\pm q^{i}))$ (i.e.,
$\sum_{i=0}^{\infty}\mu(i)(\pm 1)^{i}q^{ki}=0$), for all $i$, then $\mu=0$. In
other word, we have that for any $k\geq 0$,

$\sum_{i=0}^{\infty}\mu(2i)(q^{2})^{ki}=0,\quad\sum_{i=0}^{\infty}\mu(2i+1)(q^{2})^{ki}=0.$

Note that

$\sum_{i=0}^{\infty}\mu(2i)^{2}(q^{2})^{i^{2}}<\infty,~{}\sum_{i=0}^{\infty}\mu(2i+1)^{2}(q^{2})^{i^{2}}<\infty.$

So by the lemma 3.4, $\mu=0$. ∎

######  Corollary 4.3.

$\|\delta_{0}P_{\mathrm{sym}}^{n}-\pi_{\mathrm{sym}}\|_{tv}=\begin{cases}\frac{2q\cdot\alpha(q)}{q^{2}-1}\cdot\frac{1}{q^{n}}+O\left(\frac{1}{q^{2n}}\right),&\mbox{if
}n\text{ even}\\\
\frac{2q\cdot\alpha(q)}{(q^{2}-1)(q-1)}\cdot\frac{1}{q^{n}}+O\left(\frac{1}{q^{2n}}\right),&\mbox{otherwise}.\end{cases}$

where the implicit constant is less than $\alpha(q)^{-1}$.

###### Proof.

Similarly, we have

$\nu_{+}:=\pi_{\mathrm{sym}}-\frac{1}{2}(\pi_{\mathrm{sym}}\circ q)\in
V_{q^{-1}},\quad\nu_{-}:=\pi_{\mathrm{sym}}\circ(-q)\in V_{-q^{-1}}.$

For the $k$-moment,

$M(\pi_{\mathrm{sym}},0)=1,~{}M(\pi_{\mathrm{sym}},1)=2,~{}M(\pi_{\mathrm{sym}},2)=2+2q.$
$\langle\nu_{+},\nu_{+}\rangle_{\pi_{\mathrm{sym}}}=\frac{q-1}{2},~{}(\delta_{0})_{+}=\frac{1}{q-1}\nu_{+}.$
$\langle\nu_{-},\nu_{-}\rangle_{\pi_{\mathrm{sym}}}=2+2q,~{}(\delta_{0})_{-}=\frac{1}{2(q+1)}\nu_{-}.$

Then

$\|(\delta_{0})_{+}+(-1)^{n}(\delta_{0})_{-}\|_{tv}=\begin{cases}\frac{2q\cdot\alpha(q)}{q^{2}-1},&\mbox{if
}n\text{ even}\\\
\frac{2q\cdot\alpha(q)}{(q^{2}-1)(q-1)},&\mbox{otherwise}.\end{cases}$

∎

###  4.4. Alternating case

For alternating matrices (skew symmetric matrices), i.e. the set
${\mathrm{Mat}}_{n}^{\mathrm{alt}}({\mathbb{F}}_{q}):=\left\\{M\in{\mathrm{Mat}}_{n}({\mathbb{F}}_{q}):M^{T}=-M,\text{
and the diagonal elements of $M$ are zero}\right\\}$. It is well-known that
the rank is even (see [2, 14]). By calculation, we have

${\mathbb{P}}(\mathrm{corank}~{}M=2j+1~{}|~{}M\in{\mathrm{Mat}}_{2n+1}^{{\mathrm{alt}}})=(\delta_{0}P_{\mathrm{alt}}^{n})(j).$

and

${\mathbb{P}}(\mathrm{corank}~{}M=2j~{}|~{}M\in{\mathrm{Mat}}_{2n}^{{\mathrm{alt}}})=(\delta_{0}Q_{\mathrm{alt}}^{n})(j).$

Here

$P_{\mathrm{alt}}(i,j)=\begin{cases}q^{-4i-3},&\mbox{if }j=i+1\\\
1-q^{-4i-3}-(1-q^{-2i})(1-q^{-2i-1}),&\mbox{if }j=i\\\
(1-q^{-2i})(1-q^{-2i-1}),&\mbox{if }j=i-1\\\ 0,&\mbox{otherwise.}\end{cases}$
$Q_{\mathrm{alt}}(i,j)=\begin{cases}q^{-4i-1},&\mbox{if }j=i+1\\\
1-q^{-4i-1}-(1-q^{-2i})(1-q^{-2i+1}),&\mbox{if }j=i\\\
(1-q^{-2i})(1-q^{-2i+1}),&\mbox{if }j=i-1\\\ 0,&\mbox{otherwise.}\end{cases}$

and

$\pi_{\mathrm{alt}}(j)=\frac{\alpha(q)}{q^{2j^{2}+j}\eta_{2j+1}(q)},\quad\pi_{\mathrm{alt}}^{\prime}(j)=\frac{\alpha(q)}{q^{2j^{2}-j}\eta_{2j}(q)}.$

######  Theorem 4.4.

$P_{\mathrm{alt}}$ (resp. $Q_{\mathrm{alt}}$) is Hilbert-Schmidt on
$\ell^{2}(\pi_{\mathrm{alt}})$ (resp. $\ell^{2}(\pi^{\prime}_{\mathrm{alt}})$)
and
${\mathrm{EV}}(P_{\mathrm{alt}})={\mathrm{EV}}(Q_{\mathrm{alt}})=\left\\{q^{-2k}:k\geq
0\right\\}$.

###### Proof.

The eigenvectors of $P_{\mathrm{alt}}$ associated with eigenvalue $q^{-2k}$
can be obtained as a linear combination of vectors
$\\{\pi_{\mathrm{alt}}\circ(q^{i}):i=0,\cdots,k\\}$. Now it remains to prove
that if

$\sum_{i=0}^{\infty}\mu(i)^{2}(q^{2})^{i^{2}+i/2}<\infty\text{ and
}\sum_{i=0}^{\infty}\mu(i)(q^{2})^{ki}=0,$

then $\mu=0$. This can be directly inferred from the lemma 3.4. The proof of
$Q_{\mathrm{alt}}$ is similar. ∎

######  Corollary 4.5.

$\|\delta_{0}P_{\mathrm{alt}}^{n}-\pi_{\mathrm{alt}}\|_{tv}=\frac{2\alpha(q)}{(q-1)^{2}(q+1)}\cdot\frac{1}{q^{2n}}+O\left(\frac{1}{q^{4n}}\right),$
$\|\delta_{0}Q_{\mathrm{alt}}^{n}-\pi_{\mathrm{alt}}^{\prime}\|_{tv}=\frac{\alpha(q)q}{(q-1)(q+1)}\frac{1}{q^{2n}}+O\left(\frac{1}{q^{4n}}\right),$

where the implicit constants are less than $\eta_{1}(q)\alpha(q)^{-1}$ and
$\alpha(q)^{-1}$ respectively.

###### Proof.

For $P_{\mathrm{alt}}$, we can find
$\nu:=\pi_{\mathrm{alt}}-\frac{q}{q+1}(\pi_{\mathrm{alt}}\circ q^{2})\in
V_{q^{-2}}$ and

$(1+q^{2})(1+q^{-1})(\pi_{\mathrm{alt}}-\pi_{\mathrm{alt}}\circ
q^{2})+\pi_{\mathrm{alt}}\circ q^{4}\in V_{q^{-4}}$

For the $k$-moment,

$M(\pi_{\mathrm{alt}},0)=1,~{}M(\pi_{\mathrm{alt}},2)=1+q^{-1},~{}M(\pi_{\mathrm{alt}},4)=(1+q^{2})(1+q^{-1})q^{-1}.$

Then

$\langle\nu,\nu\rangle_{\pi_{\mathrm{alt}}}=q(q-1)(q+1)^{-1}\text{ and
}\|\nu\|_{tv}=2\nu(0)=\frac{2\alpha(q)}{(q+1)\eta_{1}(q)}.$

Then the $q^{-2}$-component of $\delta_{0}$ is

$(\delta_{0})_{q^{-2}}=q^{-1}(q-1)^{-1}\nu$

and

$\|(\delta_{0})_{q^{-2}}\|_{tv}=\frac{2\alpha(q)}{(q-1)^{2}(q+1)}.$

For $Q_{\mathrm{alt}}$, we can find
$\nu^{\prime}:=\pi_{\mathrm{alt}}-\frac{1}{q+1}(\pi_{\mathrm{alt}}^{\prime}\circ
q^{2})\in V_{q^{-2}}$ and

$\pi_{\mathrm{alt}}^{\prime}-\frac{1}{q}(\pi_{\mathrm{alt}}^{\prime}\circ
q^{2})+\frac{1}{q(q+1)(q^{2}+1)}(\pi_{\mathrm{alt}}^{\prime}\circ q^{4})\in
V_{q^{-4}}.$

For the $k$-moment,

$M(\pi_{\mathrm{alt}}^{\prime},0)=1,~{}M(\pi_{\mathrm{alt}}^{\prime},2)=q+1,~{}M(\pi_{\mathrm{alt}}^{\prime},4)=(q+1)(q^{2}+1).$

Then

$\langle\nu^{\prime},\nu^{\prime}\rangle_{\pi_{\mathrm{alt}}^{\prime}}=\frac{q(q-1)}{q+1}\text{
and }\|\nu^{\prime}\|_{tv}=\frac{\alpha(q)q}{q+1}.$

Then the $q^{-2}$-component of $\delta_{0}$ is

$(\delta_{0})_{q^{-2}}=(q-1)^{-1}\nu^{\prime}$

and

$\|(\delta_{0})_{q^{-2}}\|_{tv}=\frac{\alpha(q)q}{(q-1)(q+1)}.$

∎

###  4.5. Hermitian case

Let $q$ be a power of odd prime. Suppose that
$\theta\in{\mathbb{F}}_{q^{2}},\theta^{2}\in{\mathbb{F}}_{q}$, but
$\theta\notin{\mathbb{F}}_{q}$ (see [2]). Then any
$\alpha\in{\mathbb{F}}_{q^{2}}$ can be written as $\alpha=a+b\theta$ with
$a,b\in{\mathbb{F}}_{q}$. By the conjugate of $\alpha$, we mean
$\overline{\alpha}=a-b\theta$. For
$M=(\alpha_{ij})\in{\mathrm{Mat}}_{n}({\mathbb{F}}_{q^{2}})$, denote
$M^{\ast}=(\overline{\alpha_{ij}})^{T}$. Then $A$ is said to be Hermitian if
and only if $M^{\ast}=M$. Write
${\mathrm{Mat}}_{n}^{\mathrm{her}}({\mathbb{F}}_{q^{2}})=\left\\{M\in{\mathrm{Mat}}_{n}({\mathbb{F}}_{q^{2}}):M^{\ast}=M\right\\}$

${\mathbb{P}}(\mathrm{corank}~{}M=k~{}|~{}M\in{\mathrm{Mat}}_{n}^{{\mathrm{her}}}({\mathbb{F}}_{q^{2}}))=(\delta_{0}P_{\mathrm{her}}^{n})(k).$

Here

$P_{\mathrm{her}}(i,j)=\begin{cases}q^{-2i-1},&\mbox{if }j=i+1\\\
q^{-2i}(1-q^{-1}),&\mbox{if }j=i\\\ 1-q^{-2i},&\mbox{if }j=i-1\\\
0,&\mbox{otherwise.}\end{cases}$

and

$\pi_{\mathrm{her}}(j)=\frac{\beta(q)}{q^{j^{2}}\eta_{j}(q^{2})},\text{ where
}\beta(q)=\prod_{i=1,\text{odd}}^{\infty}(1+q^{-i})^{-1}.$

######  Theorem 4.6.

$P_{\mathrm{her}}$ is Hilbert-Schmidt on $\ell^{2}(\pi_{\mathrm{her}})$ and
${\mathrm{EV}}(P_{\mathrm{her}})=\left\\{(-q)^{-k}:k\geq 0\right\\}$.

###### Proof.

The proof is similar. We need to show

$\lim_{N\rightarrow\infty}\sum_{i=1}^{\infty}\frac{1}{q^{i^{2}}}\cdot\left|\sum_{k=0}^{N}b_{k}^{\prime}(-q)^{ki}\right|^{2}=0,\text{
where }b_{k}^{\prime}:=\frac{(-1)^{k}}{\prod_{j=1}^{k}((-q)^{j}-1)}.$

It suffices to prove

$\lim_{N\rightarrow\infty}\sum_{i=1,odd}^{\infty}\frac{1}{q^{i^{2}}}\cdot\left|\sum_{k=0}^{N}b_{k}^{\prime}(-q)^{ki}\right|^{2}=\lim_{N\rightarrow\infty}\sum_{i=1,even}^{\infty}\frac{1}{q^{i^{2}}}\cdot\left|\sum_{k=0}^{N}b_{k}^{\prime}(-q)^{ki}\right|^{2}=0$

When $i$ is odd,

${\mathrm{sign}}(b_{k}^{\prime}(-q)^{ki})\text{ is }\begin{cases}>0,&\mbox{if
}k\equiv 3,4\bmod 4\\\ <0,&\mbox{if }k\equiv 1,2\bmod 4.\end{cases}$

Note that

$\left|\frac{b_{k}^{\prime}(-q)^{ki}}{b_{k-1}^{\prime}(-q)^{(k-1)i}}\right|=\frac{q^{i}}{q^{k}-(-1)^{k}}.$

According to the property of alternating series, we have

$\sum_{i=1,odd}^{N}\frac{1}{q^{i^{2}}}\cdot\left|\sum_{k=N+1}^{\infty}b_{k}^{\prime}(-q)^{ki}\right|^{2}\ll\frac{1}{q^{(N+1)(N+2)}}\sum_{i=1,odd}^{N}\frac{1}{q^{i^{2}}}q^{2(N+1)i}\rightarrow
0\text{ as }N\rightarrow\infty.$

and

$\sum_{i=N+1,odd}^{\infty}\frac{1}{q^{i^{2}}}\cdot\left|\sum_{k=0}^{N}b_{k}^{\prime}(-q)^{ki}\right|^{2}\ll\frac{1}{q^{N}}\rightarrow
0\text{ as }N\rightarrow\infty.$

The estimation for $i$ even part is similar. ∎

######  Corollary 4.7.

$\|\delta_{0}P_{\mathrm{her}}^{n}-\pi_{\mathrm{her}}\|_{tv}=\frac{2\beta(q)}{(q+1)\alpha(q^{2})}\cdot\frac{1}{q^{n}}+O\left(\frac{1}{q^{2n}}\right),$

where the implicit constant is less than $\beta(q)^{-1}$.

###### Proof.

Similarly, $\nu:=\pi_{\mathrm{her}}\circ(-q)$ is an eigenvector associated
with $-q^{-1}$ and $\pi_{\mathrm{her}}-\frac{1}{q+1}(\pi_{\mathrm{her}}\circ
q^{2})\in V_{q^{-2}}$. By calculation,
$\langle\nu,\nu\rangle_{\pi_{\mathrm{her}}}=q+1$ and
$\|\nu\|_{tv}=\frac{2\beta(q)}{\alpha(q^{2})}$. The $-q^{-1}$-component of
$\delta_{0}$ is

$(\delta_{0})_{-q^{-1}}=\frac{1}{(q+1)}\nu\text{ and
}\|(\delta_{0})_{-q^{-1}}\|_{tv}=\frac{2\beta(q)}{(q+1)\alpha(q^{2})}.$

∎

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