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The paper develops a stochastic model of drift in human beliefs that shows that today's sheer volume of accessible information, combined with consumers' confirmation bias and natural preference to more outlying content, necessarily lead to increased polarization. The model explains the paradox of growing ideological fragmentation in the age of increased sharing. As social media, search engines, and other real-time information sharing outlets purport to facilitate access to information, a need for content filtering arises due to the ensuing information overload. In general, consumers select information that matches their individual views and values. The bias inherent in such selection is echoed by today's information curation services that maximize user engagement by filtering new content in accordance with observed consumer preferences. Consequently, individuals get exposed to increasingly narrower bands of the ideology spectrum, thus fragmenting society into increasingly ideologically isolated enclaves. We call this dynamic the paradox of information access. The model also suggests the disproportionate damage attainable with a small infusion of well-positioned misinformation. The paper describes the modeling methodology, and evaluates modeling results for different population sizes and parameter settings.